无穷级数
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数列极限
若数列$S_n$的极限值收敛于$S$，则对于任意的正实数$ε$，必有任意多项落在区间$[S - ε, S + ε]$内，且至多有限项落在区间$[S - ε, S + ε]$外。
| $\lim\limits_{n⇝∞⁺} S_n ⇝ S ≠ ∞$ | $\lim\limits_{n⇝∞⁺} S_n ⇝ S⁺ ≠ ∞⁺$ | $\lim\limits_{n⇝∞⁺} S_n ⇝ S⁻ ≠ ∞⁻$ | ||
|---|---|---|---|---|
| $\lim\limits_{n⇝∞⁺} | S_n - S | ⇝ 0⁺$ | $\lim\limits_{n⇝∞⁺} S_n - S⁺ ⇝ 0⁺$ | $\lim\limits_{n⇝∞⁺} S_n - S⁻ ⇝ 0⁻$ |
| $∀ε>0;∃N∈ℕ;∀n≥N; | S_n - S | ≤ ε$ | $∀ε>0;∃N∈ℕ;∀n≥N; 0⁺ ≤ S_n - S ≤ ε$ | $∀ε>0;∃N∈ℕ;∀n≥N; ε ≤ S_n - S ≤ 0⁻$ |
| $∀ε>0;∃N∈ℕ;∀n≥N; S - ε ≤ S_n ≤ S + ε$ | $∀ε>0;∃N∈ℕ;∀n≥N; S + 0⁺ ≤ S_n ≤ S + ε$ | $∀ε>0;∃N∈ℕ;∀n≥N; S + ε ≤ S_n ≤ S + 0⁻$ |
若数列$S_n$的极限值发散于$∞$，则对于任意的正实数$ε$，必有任意多项落在区间$[-ε, +ε]$外，且至多有限项落在区间$[-ε, +ε]$内。
| $\lim\limits_{n⇝∞⁺} S_n ⇝ ∞$ | $\lim\limits_{n⇝∞⁺} S_n ⇝ ∞⁺$ | $\lim\limits_{n⇝∞⁺} S_n ⇝ ∞⁻$ | ||
|---|---|---|---|---|
| $\lim\limits_{n⇝∞⁺} | S_n | ⇝ ∞⁺$ | ||
| $∀ε>0;∃N∈ℕ;∀n≥N; | S_n | ≥ ε$ | $∀ε>0;∃N∈ℕ;∀n≥N; S_n ≥ +ε$ | $∀ε>0;∃N∈ℕ;∀n≥N; S_n ≤ -ε$ |
| $∀ε>0;∃N∈ℕ;∀n≥N; [S_n ≤ -ε] ∨ [S_n ≥ +ε]$ |
数列$S_n$的下极限与上极限。
$\varliminf\limits_{n⇝∞⁺} S_n ≡ \mathop{\lim\inf}\limits_{n⇝∞⁺} S_n ≡ \lim\limits_{n⇝∞⁺} \inf\limits_{m≥n} S_m$
$\varlimsup\limits_{n⇝∞⁺} S_n ≡ \mathop{\lim\sup}\limits_{n⇝∞⁺} S_n ≡ \lim\limits_{n⇝∞⁺} \sup\limits_{m≥n} S_m$
若数列$S_n$中有限多项发生改变，数列的敛散性不会改变。
若数列$S_n$的极限值收敛于$S$，则其上极限与其下极限相等，反之亦然。因此若数列$S_n$的极限值存在，则此极限值唯一确定。
$\left[ \lim\limits_{n⇝∞⁺} S_n ⇝ S \right] ⇔ \left[ \varliminf\limits_{n⇝∞⁺} S_n ⇝ S ⇜ \varlimsup\limits_{n⇝∞⁺} S_n \right]$
若数列$S_n$的极限值收敛于$S$，则其任何子数列$S_{n_m}$的极限值均收敛于S，反之亦然。
$\left[ \lim\limits_{n⇝∞⁺} S_n ⇝ S \right] ⇔ \left[ \lim\limits_{m⇝∞⁺} S_{n_m} ⇝ S \right]$
若数列$S_n$的极限值收敛于$S$，则其任意无穷项$S_m$与$S_n$之差为无穷小。
| $[∀ε>0;∃N∈ℕ;∀n≥N; | S_n - S | < ε] ⇔ [∀ε>0;∃N∈ℕ;∀n,m≥N; | S_m - S_n | < ε]$ |
典例：数列$S_n = (-1)^n$的极限值不存在。
$\lim\limits_{m⇝∞⁺} S_{2·m} = \lim\limits_{m⇝∞⁺} (-1)^{2·m} ⇝ (+1) ≠ (-1) ⇜ \lim\limits_{m⇝∞⁺} (-1)^{2·m+1} = \lim\limits_{m⇝∞⁺} S_{2·m+1}$
$\varliminf\limits_{n⇝∞⁺} S_n ⇝ (-1) ≠ (+1) ⇜ \varlimsup\limits_{n⇝∞⁺} S_n$
典例：数列$S_n = \sin n$的极限值不存在。
$\varliminf\limits_{n⇝∞⁺} S_n = \varliminf\limits_{n_m⇝∞⁺}^{n_m∈\left(2·m·π-\frac{3}{4}·π,2·m·π-\frac{1}{4}·π\right)} \sin n_m < 0 < \varlimsup\limits_{n_m⇝∞⁺}^{n_m∈\left(2·m·π+\frac{1}{4}·π,2·m·π+\frac{3}{4}·π\right)} \sin n_m = \varlimsup\limits_{n⇝∞⁺} S_n$
若数列$S_n$的极限值收敛，则数列$S_n$有确界，反之不对。
| $\left[ \lim\limits_{n⇝∞⁺} S_n ⇝ S ≠ ∞ \right] ⇒ \left[ ∀n∈ℕ; | S_n | ≤ \max\left\lbrace \left | \inf S_n\right | , \left | \sup S_n \right | \right\rbrace \right]$ |
| 若数列$ | S_n | $单调递增无上确界，则数列$ | S_n | $的极限值发散于$∞⁺$。$P ⇒ Q$ |
| 若数列$ | S_n | $单调递增有上确界，则数列$ | S_n | $的极限值必定收敛。$¬Q ⇒ ¬P$ |
| $\left[ \lim\limits_{n⇝∞⁺} S_n^{↗} ⇝ ∞^{+} \right] ∨ \left[ \lim\limits_{n⇝∞⁺} S_n^{↘} ⇝ ∞^{-} \right] ⇒ \left[ \lim\limits_{n⇝∞⁺} | S_n^{⤨} | ⇝ ∞⁺ \right]$ |
| $\left[ \lim\limits_{n⇝∞⁺} | S_n^{⤨} | ≤ \mathrm{Sup} \right] ⇒ \left[ \lim\limits_{n⇝∞⁺} S_n^{↗} ⇝ \sup S_n \right] ∧ \left[ \lim\limits_{n⇝∞⁺} S_n^{↘} ⇝ \inf S_n \right]$ |
典例：若数列$0 ≤ S_{n+m} ≤ S_{n} + S_{m}$，则数列$\dfrac{S_n}{n}$的极限值收敛。
| $⇓$ | $n \mathop{=}\limits_{0≤r≤m} m · q + r$ | $⇒$ | $\dfrac{1}{m} = \dfrac{q}{n} + \dfrac{r}{n · m}$ |
|---|---|---|---|
| $⇓$ | $\dfrac{S_n}{n} = \dfrac{S_{m·q+r} }{n} = \dfrac{q · S_{m} }{n} + \dfrac{S_{r} }{n} \mathop{≤}\limits^{0≤S_{m} } \dfrac{S_{m} }{m} + \dfrac{S_{r} }{n}$ | $⇐$ | $\dfrac{1}{m} ≥ \dfrac{q}{n}$ |
| $⇓$ | $\varlimsup\limits_{n⇝∞^{+} } \dfrac{S_{n} }{n} ≤ \varlimsup\limits_{n⇝∞^{+} } \left( \dfrac{S_{m} }{m} + \dfrac{S_{r} }{n} \right) = \dfrac{S_{m} }{m} + 0 = \dfrac{S_{m} }{m}$ | $⇒$ | $\varlimsup\limits_{n⇝∞^{+} } \dfrac{S_{n} }{n} ≤ \varliminf\limits_{m⇝∞^{+} } \dfrac{S_{m} }{m}$ |
| $⇓$ | $\varlimsup\limits_{n⇝∞^{+} } \dfrac{S_{n} }{n} = \lim\limits_{n⇝∞^{+} } \dfrac{S_{n} }{n} = \varliminf\limits_{n⇝∞^{+} } \dfrac{S_{n} }{n}$ |
数列极限的运算性质
$\left[ \lim\limits_{n⇝∞⁺} S_n ⇝ S < T ⇜ \lim\limits_{n⇝∞⁺} T_n \right] ⇒ \left[ ∃N∈ℕ;∀n≥N; S_n < \dfrac{S + T}{2} < T_n \right]$
$\left[ ∃N∈ℕ;∀n≥N; S_n ≤ R_n ≤ T_n \right] ⇒ \left[ \lim\limits_{n⇝∞⁺} S_n ≤ \lim\limits_{n⇝∞⁺} R_n ≤ \lim\limits_{n⇝∞⁺} T_n \right]$
| $⇓$ | $\lim\limits_{n⇝∞⁺} S_n ⇝ S < T ⇜ \lim\limits_{n⇝∞⁺} T_n$ |
|---|---|
| $⇓$ | $∀ε=\dfrac{T - S}{2};∃N∈ℕ;∀n≥N; S_n < S + ε = S + \dfrac{T - S}{2} = \dfrac{S + T}{2} = T - \dfrac{T - S}{2} = T - ε < T_n$ |
| $⇓$ | $∃N∈ℕ;∀n≥N; S_n < \dfrac{S + T}{2} < T_n$ |
数列极限的运算性质。若$\lim\limits_{n⇝∞⁺} S_n ⇝ S$，且 $\lim\limits_{n⇝∞⁺} T_n ⇝ T$。
$\lim\limits_{n⇝∞⁺} [ S_n + T_n ] = \lim\limits_{n⇝∞⁺} S_n + \lim\limits_{n⇝∞⁺} T_n ⇝ S + T$
$\lim\limits_{n⇝∞⁺} [ S_n - T_n ] = \lim\limits_{n⇝∞⁺} S_n - \lim\limits_{n⇝∞⁺} T_n ⇝ S - T$
$\lim\limits_{n⇝∞⁺} [ S_n · T_n ] = \lim\limits_{n⇝∞⁺} S_n · \lim\limits_{n⇝∞⁺} T_n ⇝ S · T$
$\lim\limits_{n⇝∞⁺} \dfrac{S_n}{T_n} = \dfrac{\lim\limits_{n⇝∞⁺} S_n}{\lim\limits_{n⇝∞⁺} T_n} \mathop{⇝}\limits_{T≠0} \dfrac{S}{T}$
$\lim\limits_{n⇝∞⁺} [ \mathrm{Con} · S_n ] = \mathrm{Con} · \lim\limits_{n⇝∞⁺} S_n ⇝ \mathrm{Con} · S$
| $⇓$ | $[ ∀ε_1>0;∃N_1∈ℕ;∀n≥N_1; | S_n - S | < ε_1 ] ∧ [∀ε_2>0;∃N_2∈ℕ;∀n≥N_2; | T_n - T | < ε_2]$ | ||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $⇓$ | $∀ε>0;∃N=\max\lbrace N_1,N_2 \rbrace;∀n≥N; | (S_n + T_n) - (S + T) | < | S_n - S | + | T_n - T | < ε_1 + ε_2 = ε$ | ||||||||||||||||||||||||||
| $⇓$ | $∀ε>0;∃N=\max\lbrace N_1,N_2 \rbrace;∀n≥N; | (S_n - T_n) - (S - T) | < | S_n - S | + | T_n - T | < ε_1 + ε_2 = ε$ | ||||||||||||||||||||||||||
| $⇓$ | $∀ε>0;∃N=\max\lbrace N_1,N_2 \rbrace;∀n≥N; | S_n · T_n - S · T | = | (S_n · T_n - S_n · T) + (S_n · T - S · T) | ≤ | S_n | · | T_n - T | + | S_n - S | · | T | ≤ \mathrm{Sup} · ε_2 + ε_1 · | T | = ε$ | ||||||||||||||||||
| $⇓$ | $∀ε>0;∃N=\max\lbrace N_1,N_2 \rbrace;∀n≥N; \left | \dfrac{S_n}{T_n} - \dfrac{S}{T} \right | = \dfrac{ | S_n · T - S · T_n | }{ | T_n | · | T | } = \dfrac{ | (S_n · T - S · T) + (S · T - S · T_n) | }{ | T_n | · | T | } = \dfrac{ | S_n - S | · | T | + | S | · | T - T_n | }{ | T_n | · | T | } ≤ \dfrac{ε_1 · | T | + | S | · ε_2}{\mathrm{Inf} · | T | } = ε$ |
无穷级数
无穷级数的部分和
$S_{n+1} \mathop{≡≡}\limits^{S_0≡0} \sum\limits_{i=0}^n s_i$
无穷级数的极限
$\lim\limits_{n ⇝ ∞⁺} S_n ≡ \lim\limits_{n ⇝ ∞⁺} \sum\limits_{i=0}^{n} s_i ≡ \sum\limits_{i=0}^{∞⁺} s_i$
无穷级数的下极限与上极限
$\varliminf\limits_{n ⇝ ∞⁺} S_n ≡ \mathop{\lim\inf}\limits_{n ⇝ ∞⁺} S_n ≡ \lim\limits_{n ⇝ ∞⁺}\inf\limits_{m ≥ n} S_m ≡ \lim\limits_{n ⇝ ∞⁺}\inf\limits_{m ≥ n} \sum\limits_{i = 0}^m s_i$
$\varlimsup\limits_{n⇝∞⁺} S_n ≡ \mathop{\lim\sup}\limits_{n⇝∞⁺} S_n ≡ \lim\limits_{n⇝∞⁺}\sup\limits_{m≥n} S_m ≡ \lim\limits_{n⇝∞⁺}\sup\limits_{m≥n} \sum\limits_{i=0}^m s_i$
$\dfrac{1}{\varlimsup\limits_{n⇝∞⁺} T_n} = \varliminf\limits_{n⇝∞⁺} \dfrac{1}{T_n} = \varliminf\limits_{n⇝∞⁺} S_n ≤ \varlimsup\limits_{n⇝∞⁺} S_n = \varlimsup\limits_{n⇝∞⁺} \dfrac{1}{T_n} = \dfrac{1}{\varliminf\limits_{n⇝∞⁺} T_n}$
典例：无穷级数的下极限与上极限
$\varliminf\limits_{n ⇝ ∞⁺} \dfrac{1}{n} ⇝ 0, \varlimsup\limits_{n ⇝ ∞⁺} \dfrac{1}{n} ⇝ 0$
$\varliminf\limits_{n ⇝ ∞⁺} \sin\dfrac{n · \pi}{4} ⇝ -1, \varlimsup\limits_{n ⇝ ∞⁺} \sin\dfrac{n · \pi}{4} ⇝ +1$
无穷级数的极限之等价表述形式
| $∀ε>0;∃N∈ℕ;∀n≥N; | S_n - S | < ε$ |
| $∀ε>0;∃N∈ℕ;∀n≥N; \left | \sum\limits_{i=0}^{n} s_i - S \right | < ε$ |
无穷级数收敛
| $\lim\limits_{n ⇝ ∞⁺} S_n ⇝ S$ 也即 $\lim\limits_{n ⇝ ∞⁺} | S_n - S | ⇝ 0$ |
| $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S$ 也即 $\lim\limits_{n⇝∞⁺} \left | \sum\limits_{i=0}^n s_i - S \right | ⇝ 0$ |
$\varliminf\limits_{n ⇝ ∞⁺} S_n = \lim\limits_{n ⇝ ∞⁺} S_n = \varlimsup\limits_{n ⇝ ∞⁺} S_n$
无穷级数发散
| $\lim\limits_{n ⇝ ∞⁺} S_n \not⇝ S$ 也即 $\lim\limits_{n ⇝ ∞⁺} | S_n - S | \not⇝ 0$ |
| $\sum\limits_{i=0}^{∞⁺} s_i \not⇝ S$ 也即 $\lim\limits_{n⇝∞⁺} \left | \sum\limits_{i=0}^n s_i - S \right | \not⇝ 0$ |
$\varliminf\limits_{n ⇝ ∞⁺} S_n ≠ \varlimsup\limits_{n ⇝ ∞⁺} S_n$
无穷级数收敛的性质
若无穷级数收敛，则其部分和的绝对值收敛。
| $\lim\limits_{n⇝∞⁺} S_n ⇝ S ⇒ \lim\limits_{n⇝∞⁺} | S_n | ⇝ | S | $ |
| $[ ∀ε>0;∃N∈ℕ;∀n≥N; | S_n - S | < ε ] ⇒ [ ∀ε>0;∃N∈ℕ;∀n≥N; | S_n | - | S | ≤ | S_n - S | < ε]$ |
若无穷级数收敛，则其通项收敛于零。
$\lim\limits_{n ⇝ ∞⁺} S_n ⇝ S ⇒ \lim\limits_{i ⇝ ∞⁺} s_i ⇝ 0$
若通项不收敛于零，则无穷级数发散。
$\lim\limits_{i ⇝ ∞⁺} s_i \not⇝ 0 ⇒ \lim\limits_{n ⇝ ∞⁺} S_n \not⇝ S$
将无穷级数的有限多项改变，其敛散性不变。
$\sum\limits_{i = n}^{∞⁺} s_i ⇝ S ⇔ \sum\limits_{i = m}^{∞⁺} s_i ⇝ T$
若无穷级数收敛，再将通项保序归组，则其收敛性不变。
$\sum\limits_{i=0}^{∞⁺} s_i ⇝ S ⇒ \sum\limits_{i=0}^{∞⁺} (s_{i_0} + ⋯ + s_{i_j}) ⇝ S$
若无穷级数各组内的通项正负性相同，则其敛散性不变。
$\sum\limits_{i=0}^{∞⁺} s_i ⇝ S ⇔ \sum\limits_{i=0}^{∞⁺} (s_{i_0} + ⋯ + s_{i_j}) ⇝ S$
特例：无穷级数$\sum\limits_{i=0}^{∞⁺} (-1)^i \not⇝ S$。
$\lim\limits_{i ⇝ ∞⁺} S_{2 · i - 1} ⇝ -1, \lim\limits_{i ⇝ ∞⁺} S_{2 · i} ⇝ 0$
无穷级数的数列审敛法
若无穷级数收敛，则其部分和的极限之差为无穷小。
| $[∀ε>0;∃N∈ℕ;∀n≥N; | S_n - S | < ε] ⇔ [∀ε>0;∃N∈ℕ;∀n≥N;∀m>N; | S_m - S_n | < ε]$ |
| $[∀ε>0;∃N∈ℕ;∀n≥N; \left | \sum\limits_{i=0}^n s_i - S \right | < ε] ⇔ [∀ε>0;∃N∈ℕ;∀n≥N;∀m>n; \left | \sum\limits_{i=n}^m s_i \right | < ε]$ |
| $[∀ε>0;∃N∈ℕ; n ≥ N ⇒ | S_n - S | < ε] ⇔ [∀ε>0;∃N∈ℕ; n ≥ N ∧ m > N ⇒ | S_m - S_n | < ε]$ |
| $[∀ε>0;∃N∈ℕ; n ≥ N ⇒ \left | \sum\limits_{i=0}^n s_i - S \right | < ε] ⇔ [∀ε>0;∃N∈ℕ; n ≥ N ∧ m > n ⇒ \left | \sum\limits_{i=n}^m s_i \right | < ε]$ |
若无穷级数发散，则其部分和的极限之差非无穷小。
| $[∃ε>0;∀N∈ℕ;∃n≥N; | S_n - S | ≥ ε] ⇔ [∃ε>0;∀N∈ℕ;∃n≥N;∃m>N; | S_m - S_n | ≥ ε]$ |
| $[∃ε>0;∀N∈ℕ;∃n≥N; \left | \sum\limits_{i=0}^n s_i - S \right | ≥ ε] ⇔ [∃ε>0;∀N∈ℕ;∃n≥N;∃m>n; \left | \sum\limits_{i=n}^m s_i \right | ≥ ε$ |
| $[∃ε>0;∀N∈ℕ; n ≥ N ∧ | S_n - S | ≥ ε] ⇔ [∃ε>0;∀N∈ℕ; n ≥ N ∧ m > N ∧ | S_m - S_n | ≥ ε]$ |
| $[∃ε>0;∀N∈ℕ; n ≥ N ∧ \left | \sum\limits_{i=0}^n s_i - S \right | ≥ ε] ⇔ [∃ε>0;∀N∈ℕ; n ≥ N ∧ m > n ∧ \left | \sum\limits_{i=n}^m s_i \right | ≥ ε]$ |
必要性证明，根据三角不等关系式。
| $⇓$ | $∀ε_1>0;∃N∈ℕ;∀n≥N; \lvert S_n - S \rvert < ε_1] ∧ [∀ε_2>0;∃N∈ℕ;∀m>N; \lvert S_m - S \rvert < ε_2$ |
|---|---|
| $⇓$ | $∀ε=ε_1 + ε_2;∃N∈ℕ;∀n≥N;∀m>N; \lvert S_m - S_n \rvert ≤ \lvert S_m - S \rvert + \lvert S_n - S \rvert < ε_1 + ε_2 = ε$ |
| $⇓$ | $∀ε>0;∃N∈ℕ;∀n≥N;∀m>N; \lvert S_m - S_n \rvert < ε$ |
充分性证明，根据上下确界关系式。
| $⇓$ | $∀ε>0;∃N∈ℕ;∀n≥N;∀m>N; \lvert S_m - S_n \rvert < ε$ |
|---|---|
| $⇓$ | $∃S;∀ε=ε_1+ε_2;∃N∈ℕ;∀n≥N;∀m>N; S - ε_1 < S_m, S_n < S + ε_2$ |
| $⇓$ | $∀ε>0;∃N∈ℕ;∀n≥N;\lvert S_n - S \rvert < ε$ |
无穷级数的分部审敛法
错位加和公式。
$\sum\limits_{i=n}^m s_i · t_i \mathop{===}\limits^{S_{-1}≡0} \sum\limits_{i=n}^m (S_i - S_{i-1}) · t_i = S_m · t_m + \sum\limits_{i=n}^{m-1} S_i · t_i - \sum\limits_{i=n}^{m-1} S_i · t_{i+1} - S_{n-1} · t_n = (S_m · t_m - S_{n-1} · t_n) + \sum\limits_{i=n}^{m-1} S_i · (t_i - t_{i+1}) $
| $\left | \sum\limits_{i=n}^m s_i · t_i \right | ≤ | S_m · t_m | + | S_{n-1} · t_n | + \left | \sum\limits_{i=n}^{m-1} S_i · (t_i - t_{i+1}) \right | ≤ | S_m | · | t_m | + | S_{n-1} | · | t_n | + \sum\limits_{i=n}^{m-1} | S_i | · | t_i - t_{i+1} | $ |
| $⇓$ | $[ ∃N∈ℕ;∀i>N; | S_i | ≤ \mathrm{Sup.} ∧ t_i ≥ t_{i+1} ] ⊕ [ ∃N∈ℕ;∀i>N; | S_i | ≤ \mathrm{Sup.} ∧ t_i ≤ t_{i+1} ]$ | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $⇓$ | $\left | \sum\limits_{i=n}^m s_i · t_i \right | ≤ \mathrm{Sup.} · \left[ | t_m | + | t_n | + \left | \sum\limits_{i=n}^{m-1} (t_i - t_{i+1}) \right | \right] ≤ \mathrm{Sup.} · 2 · \left[ | t_m | + | t_n | \right]$ | ||
| $⇓$ | $\varlimsup\limits_{i ⇝ ∞⁺} | S_i | ≤ \mathrm{Sup.} ∧ t_i ⪌ t_{i+1} ∧ \lim\limits_{i ⇝ ∞⁺} t_i ⇝ 0$ | $⇒$ | $∀ε>0;∃N∈ℕ;∀n≥N;∀m≥n; \left | \sum\limits_{i=n}^m s_i · t_i \right | < \mathrm{Sup.} · 2 · (ε_1 + ε_2) = ε$ | ||||||||
| $⇓$ | $\varlimsup\limits_{i ⇝ ∞⁺} | S_i | ≤ \mathrm{Sup.} ∧ t_i ⪌ t_{i+1} ∧ \lim\limits_{i ⇝ ∞⁺} t_i ⇝ 0$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i · t_i ⇝ ST$ | ||||||||||
| $⇓$ | $\lim\limits_{i ⇝ ∞⁺} S_i ⇝ S ∧ t_i ⪌ t_{i+1} ∧ \lim\limits_{i ⇝ ∞⁺} t_i ⇝ t$ | $\mathop{⇒}\limits_{\lim\limits_{i⇝∞⁺} (t_i - t) ⇝ 0}$ | $\sum\limits_{i=0}^{∞⁺} s_i · t_i = t · \sum\limits_{i=0}^{∞⁺} s_i + \sum\limits_{i=0}^{∞⁺} [ s_i · (t_i - t) ] ⇝ ST$ | ||||||||||||
| $⇓$ | $\lim\limits_{i ⇝ ∞⁺} S_i ⇝ S ∧ t_i ⪌ t_{i+1} ∧ \varlimsup\limits_{i⇝∞⁺} | t_i | ≤ \mathrm{Sup.}$ | $\mathop{⇒}\limits_{\lim\limits_{i⇝∞⁺} t_i ⇝ t}$ | $\sum\limits_{i=0}^{∞⁺} s_i · t_i ⇝ ST$ |
若部分和$\sum\limits_{i=0}^n s_i$有确界，且$t_i$单调趋于零，则无穷级数$\sum\limits_{i=0}^{∞⁺} s_i · t_i$收敛。
若无穷级数$\sum\limits_{i=0}^{∞⁺} s_i$收敛，且$t_i$单调有确界，则无穷级数$\sum\limits_{i=0}^{∞⁺} s_i · t_i$收敛。
特例：若$t_i$单调趋于零，则交错级数$\sum\limits_{i=0}^{∞⁺} (-1)^i · t_i$收敛。
无穷级数的绝对收敛与条件收敛
若无穷级数绝对收敛，则其本身必定收敛。
| $\sum\limits_{i=0}^{∞⁺} | s_i | ⇝ S_{ | } ⇒ \sum\limits_{i=0}^{∞⁺} s_i ⇝ S$ |
| $\left[ ∀ε>0;∃N∈ℕ;∀n≥N;∀m>n; \left | \sum\limits_{i=n}^m | s_i | \right | < ε \right] ⇒ \left[ ∀ε>0;∃N∈ℕ;∀n≥N;∀m>n; \left | \sum\limits_{i=n}^m s_i \right | ≤ \left | \sum\limits_{i=n}^{m} | s_i | \right | < ε \right]$ |
若无穷级数本身收敛，则其未必绝对收敛。
| $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S \not⇒ \sum\limits_{i=0}^{∞⁺} | s_i | ⇝ S_{ | }$ |
| $\left[ ∀ε>0;∃N∈ℕ;∀n≥N;∀m>n; \left | \sum\limits_{i=n}^m s_i \right | < ε \right] \not⇒ \left[ ∀ε>0;∃N∈ℕ;∀n≥N;∀m>n; \left | \sum\limits_{i=n}^m | s_i | \right | < ε \right]$ |
无穷级数绝对收敛与条件收敛的性质
| 无穷级数的正数项$+s_i^+ ≡ +\dfrac{1}{2} · ( | s_i | + s_i)$，无穷级数的负数项$-s_i^- ≡ -\dfrac{1}{2} · ( | s_i | - s_i)$。 |
| 无穷级数的一般项$s_i = s_i^+ - s_i^-$，无穷级数的绝对项$ | s_i | = s_i^+ + s_i^-$。 |
若无穷级数绝对收敛，则其正数项级数与负数项级数均收敛，反之亦然。
| $\sum\limits_{i=0}^{∞⁺} | s_i | = \sum\limits_{i=0}^{∞⁺} (s_i^+ + s_i^-) ⇝ S_{ | } ⇔ \left[ \sum\limits_{i=0}^{∞⁺} s_i^+ ⇝ S_{+} \right] ∧ \left[ \sum\limits_{i=0}^{∞⁺} s^- ⇝ S_- \right]$ |
若无穷级数条件收敛，则其正数项级数与负数项级数均发散，反之不对。
$\sum\limits_{i=0}^{∞⁺} s_i = \sum\limits_{i=0}^{∞⁺} (s_i^+ - s_i^-) ⇝ S ⇒ \left[ \sum\limits_{i=0}^{∞⁺} s_i^+ \not⇝ S_+ \right] ∧ \left[ \sum\limits_{i=0}^{∞⁺} s_i^- \not⇝ S_- \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} s_i^+ ⇝ ∞⁺ \right] ∧ \left[ \sum\limits_{i=0}^{∞⁺} s_i^- ⇝ ∞⁺ \right]$
| $⇓$ | $\sum\limits_{i=0}^{∞⁺} | s_i | ⇝ S_{ | } ⇒ \sum\limits_{i=0}^{∞⁺} s_i^+ ⇝ S_+$ | $⇐$ | $0 ≤ s_i^+ ≤ | s_i | $ | |||
|---|---|---|---|---|---|---|---|---|---|---|---|
| $⇓$ | $\sum\limits_{i=0}^{∞⁺} | s_i | ⇝ S_{ | } ⇒ \sum\limits_{i=0}^{∞⁺} s^- ⇝ S_-$ | $⇐$ | $0 ≤ s_i^- ≤ | s_i | $ | |||
| $⇓$ | $\sum\limits_{i=0}^{∞⁺} | s_i | ⇝ S_{ | } ⇐ \sum\limits_{i=0}^{∞⁺} s_i^+ ⇝ S_+ ∧ \sum\limits_{i=0}^{∞⁺} s_i^- ⇝ S_-$ | $⇐$ | $\sum\limits_{i=0}^{∞⁺} | s_i | = \sum\limits_{i=0}^{∞⁺} s_i^+ + \sum\limits_{i=0}^{∞⁺} s_i^-$ | |||
| $⇓$ | $\sum\limits_{i=0}^{∞⁺} | s_i | ⇝ S_{ | } ⇔ \sum\limits_{i=0}^{∞⁺} s_i^+ ⇝ S_+ ∧ \sum\limits_{i=0}^{∞⁺} s_i^- ⇝ S_-$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} | s_i | \not⇝ S_{ | } ⇒ \left[ \sum\limits_{i=0}^{∞⁺} s_i^+ \not⇝ S_+ ∨ \sum\limits_{i=0}^{∞⁺} s_i^- \not⇝ S_- \right]$ | ||
| $⇓$ | $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S ⇒ \left[ \sum\limits_{i=0}^{∞⁺} s_i^+ ⇝ S_+ ∧ \sum\limits_{i=0}^{∞⁺} s_i^- ⇝ S_- \right] ⊕ \left[ \sum\limits_{i=0}^{∞⁺} s_i^+ \not⇝ S_+ ∧ \sum\limits_{i=0}^{∞⁺} s_i^- \not⇝ S_- \right]$ | $⇐$ | $\sum\limits_{i=0}^{∞⁺} s_i = \sum\limits_{i=0}^{∞⁺} s_i^+ - \sum\limits_{i=0}^{∞⁺} s_i^-$ | ||||||||
| $⇓$ | $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S ∧ \sum\limits_{i=0}^{∞⁺} | s_i | \not⇝ S_{ | } ⇒ \left[ \sum\limits_{i=0}^{∞⁺} s_i^+ \not⇝ S_+ ∧ \sum\limits_{i=0}^{∞⁺} s_i^- \not⇝ S_- \right]$ | |||||||
| $⇓$ | $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S ∧ \sum\limits_{i=0}^{∞⁺} | s_i | \not⇝ S_{ | } ⇒ \left[ \sum\limits_{i=0}^{∞⁺} s_i^+ ⇝ ∞⁺ ∧ \sum\limits_{i=0}^{∞⁺} s_i^- ⇝ ∞⁺ \right]$ | $⇐$ | $0 ≤ s_i^+ ∧ 0 ≤ s_i^-$ |
若无穷级数绝对收敛，则交换任意多项的次序，其极限不变并且保持绝对收敛。
若无穷级数条件收敛，则交换任意多项的次序，可使其趋近于任意给定的极限。
| $⇓$ | $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S ⇒ \lim\limits_{i⇝∞⁺} s_i ⇝ 0$ | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| $⇓$ | $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S ∧ \sum\limits_{i=0}^{∞⁺} | s_i | \not⇝ S_{ | } ⇒ \sum\limits_{i=0}^{∞⁺} s_i^+ ⇝ ∞⁺ ∧ \sum\limits_{i=0}^{∞⁺} s_i^- ⇝ ∞⁺$ | ||||||
| $⇓$ | $∃n_0^+∈ℕ;∃n_0^-∈ℕ;∃n_0≤\max(n_0^+, n_0^-);$ | $\left | \left( \sum\limits_{j=0}^{n_0^+} s_j^+ - \sum\limits_{j=0}^{n_0^-} s_j^- \right) - T \right | < | s_{n_0} | $ | ||||
| $⇓$ | $∃n_1^+∈ℕ;∃n_1^-∈ℕ;∃n_1≤\max(n_1^+, n_1^-);$ | $\left | \left( \sum\limits_{j=0}^{n_0^+} s_j^+ - \sum\limits_{j=0}^{n_0^-} s_j^- \right) + \left( \sum\limits_{j=n_0^+ + 1}^{n_1^+} s_j^+ - \sum\limits_{j=n_0^- + 1}^{n_1^-} s_j^- \right) - T \right | < | s_{n_1} | < | s_{n_0} | $ | ||
| $⇓$ | $···$ | $···$ | ||||||||
| $⇓$ | $\left | \left( \sum\limits_{j=0}^{n_0^+} s_j^+ - \sum\limits_{j=0}^{n_0^-} s_j^- \right) + \left( \sum\limits_{j=n_0^+ + 1}^{n_1^+} s_j^+ - \sum\limits_{j=n_0^- + 1}^{n_1^-} s_j^- \right) + ··· + \left( \sum\limits_{j=n_{m-1}^+ + 1}^{n_m^+} s_j^+ - \sum\limits_{j=n_{m-1}^- + 1}^{n_m^-} s_j^- \right) - T \right | < ε = | s_{n_m} | < ··· < | s_{n_1} | < | s_{n_0} | $ | |
| $⇓$ | $\left( \sum\limits_{j=0}^{∞⁺} s_j^+ - \sum\limits_{j=0}^{∞⁺} s_j^- \right) = \sum\limits_{j=0}^{∞⁺} s_j ⇝ T$ |
特例：无穷级数$\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} \dfrac{(-1)^i}{i} ⇝ S$条件收敛非绝对收敛，交换无穷多项的次序可使其值发生改变。
$\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i^+ = \sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} \dfrac{1}{2 · i} ⇝ ∞⁺, \sum\limits_{i=0}^{∞⁺} s_i^- = \sum\limits_{i=0}^{∞⁺} = \dfrac{1}{2 · i + 1} ⇝ ∞⁺$
新无穷级数$- \dfrac{1}{1} + \sum\limits_{i=1}^{∞⁺} \left( - \dfrac{1}{4 · i - 1} + \dfrac{1}{2 · i} - \dfrac{1}{4 · i + 1} \right) ⇝ \dfrac{3}{2} · S$，由原无穷级数$\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} \dfrac{(-1)^i}{i} ⇝ S$，交换无穷多项的次序得到，两者并非等值。
| $S = $ | $- \dfrac{1}{1}$ | $+ \dfrac{1}{2}$ | $- \dfrac{1}{3}$ | $+ \dfrac{1}{4}$ | $- \dfrac{1}{5}$ | $+ \dfrac{1}{6}$ | $- \dfrac{1}{7}$ | $+ \dfrac{1}{8}$ | $- \dfrac{1}{9}$ | $+ \dfrac{1}{10}$ | $- \dfrac{1}{11}$ | $+ \dfrac{1}{12}$ | $- \dfrac{1}{13}$ | $+ \dfrac{1}{14}$ | $- \dfrac{1}{15}$ | $+ \dfrac{1}{16}$ | $- \dfrac{1}{17}$ | $+ \dfrac{1}{18}$ | $- \dfrac{1}{19}$ | $+ \dfrac{1}{20}$ | $-\dfrac{1}{21}$ | $+ \dfrac{1}{22}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $+) \dfrac{1}{2} · S = $ | $- \dfrac{1}{2}$ | $+ \dfrac{1}{4}$ | $- \dfrac{1}{6}$ | $+ \dfrac{1}{8}$ | $- \dfrac{1}{10}$ | $+ \dfrac{1}{12}$ | $- \dfrac{1}{14}$ | $+ \dfrac{1}{16}$ | $- \dfrac{1}{18}$ | $+ \dfrac{1}{20}$ | $- \dfrac{1}{22}$ | |||||||||||
| $\dfrac{3}{2} · S = $ | $- \dfrac{1}{1}$ | $- \dfrac{1}{3}$ | $+ \dfrac{1}{2}$ | $- \dfrac{1}{5}$ | $- \dfrac{1}{7}$ | $+ \dfrac{1}{4}$ | $- \dfrac{1}{9}$ | $- \dfrac{1}{11}$ | $+ \dfrac{1}{6}$ | $- \dfrac{1}{13}$ | $- \dfrac{1}{15}$ | $+ \dfrac{1}{8}$ | $- \dfrac{1}{17}$ | $- \dfrac{1}{19}$ | $+ \dfrac{1}{10}$ | $-\dfrac{1}{21}$ |
正项级数收敛的性质
若正项级数收敛，则正项级数有上确界，反之亦然。
$\sum\limits_{i=0}^{∞⁺} s_i ⇝ S ⇔ \sum\limits_{i=0}^{∞⁺} s_i ≤ \mathrm{Sup.}$
若正项级数发散，则正项级数无上确界，反之亦然。
$\sum\limits_{i=0}^{∞⁺} s_i \not⇝ S ⇔ \sum\limits_{i=0}^{∞⁺} s_i \not≤ \mathrm{Sup.}$
正项级数的比较审敛法
| $⇑$ | $[ ∃N∈ℕ;∀i≥N; 0 ≤ s_i ≤ t_i ] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} t_i ⇝ T ⇒ \sum\limits_{i=0}^{∞⁺} s_i ⇝ S \right]$ | $[ ∃N∈ℕ;∀i≥N; s_i ≥ t_i ≥ 0 ] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} t_i \not⇝ T ⇒ \sum\limits_{i=0}^{∞⁺} s_i \not⇝ S\right]$ | |
|---|---|---|---|
| $⇑$ | $∃N∈ℕ;∀i≥N; \dfrac{s_i}{t_i} ≤ 1$ | $⇒$ | $\dfrac{s_i}{t_i} ≤ 1 + o (1)$ |
| $⇑$ | $∃N∈ℕ;∀i≥N; \dfrac{s_i}{t_i} = γ_i$ | $⇒$ | $γ_i = \dfrac{s_i}{t_i} + o (1)$ |
| $⇑$ | $\dfrac{s_i}{t_i} = γ_i ≤ 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} t_i ⇝ T ⇒ \sum\limits_{i=0}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\dfrac{s_i}{t_i} =γ_i > 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i \not⇝ S ⇒ \sum\limits_{i=0}^{∞⁺} t_i \not⇝ T$ |
| $⇑$ | $\varlimsup\limits_{i ⇝ ∞⁺} \dfrac{s_i}{t_i} = \varlimsup\limits_{i ⇝ ∞⁺} γ_i ⇝ γ < 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} t_i ⇝ T ⇒ \sum\limits_{i=0}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\varlimsup\limits_{i ⇝ ∞⁺} \dfrac{s_i}{t_i} = \varlimsup\limits_{i ⇝ ∞⁺} γ_i ⇝ γ > 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i \not⇝ S ⇒ \sum\limits_{i=0}^{∞⁺} t_i \not⇝ T$ |
特例：正项级数$\sum\limits_{i=0}^{∞⁺} f(α + i)$与广义积分$\int\limits_α^{∞⁺} f(x) \mathrm{d}x$的敛散性相同，$f(x)$为区间$[α, ∞⁺)$上非负单调递减的连续函数。
$\left[ \sum\limits_{i=0}^{∞⁺} f(α + i) ⇝ S \right] ⇔ \left[ \int\limits_α^{∞⁺} f(x) \mathrm{d}x ⇝ I \right]$
| $⇓$ | $\sum\limits_{i = 0}^{∞⁺} f(α + i + 1) = \sum\limits_{i = 0}^{∞⁺} \int\limits_{α + i}^{α + i + 1} f(α + i + 1) \mathrm{d}x ≤ \sum\limits_{i = 0}^{∞⁺} \int\limits_{α + i}^{α + i + 1} f(x) \mathrm{d}x ≤ \sum\limits_{i = 0}^{∞⁺} \int\limits_{α + i}^{α + i + 1} f(α + i) \mathrm{d}x = \sum\limits_{i = 0}^{∞⁺} f(α + i)$ |
|---|---|
| $⇓$ | $\sum\limits_{i=1}^{∞⁺} f(α + i) = \sum\limits_{i = 0}^{∞⁺} f(α + i + 1) ≤ \int\limits_α^{∞⁺} f(x) \mathrm{d}x ≤ \sum\limits_{i = 0}^{∞⁺} f(α + i)$ |
特例：正项级数$\sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} \dfrac{1}{i^α · \ln^β i · \ln^γ \ln i}$收敛，仅当$α > 1$，以及$α = 1, β > 1$，以及$α = 1, β = 1, γ > $1。
| $⇓$ | $f(x) ≡ \dfrac{1}{x^α · \ln^β x · \ln^γ \ln x}$ | $⇒$ | $x_1 > x_2 ⇒ f(x_1) > f(x_2)$ |
|---|---|---|---|
| $⇓$ | $s_i = \dfrac{1}{i^α · \ln^β i · \ln^γ \ln i} = \dfrac{1}{i^{1 + ε_1} } · \dfrac{1}{i^{ε_2} · \ln^β i · \ln^γ \ln i}$ | $⇐$ | $α > 1 ⇒ α ≡ 1 + ε_1 + ε_2$ |
| $⇓$ | $α > 1 ⇒ s_i < \dfrac{1}{i^{1 + ε_1} }$ | $⇐$ | $\lim\limits_{i ⇝ ∞⁺} \dfrac{1}{i^{ε_2} · \ln^β i · \ln^γ \ln i} ⇝ 0$ |
| $⇓$ | $α > 1 ⇒ \sum\limits_{i=2}^{∞⁺} s_i ⇝ S$ | $⇐$ | $\int\limits_{i=2}^{∞⁺} \dfrac{1}{i^{1 + ε_1} } \mathrm{d}i ⇝ I$ |
| $⇓$ | $s_i = \dfrac{1}{i^α · \ln^β i · \ln^γ \ln i} = \dfrac{1}{i^{1 - ε_1} } · \dfrac{i^{ ε_2} }{\ln^β i · \ln^γ \ln i}$ | $⇐$ | $α < 1 ⇒ α ≡ 1 - ε_1 - ε_2$ |
| $⇓$ | $α < 1 ⇒ s_i > \dfrac{1}{i^{1 - ε_1} }$ | $⇐$ | $\lim\limits_{i ⇝ ∞⁺} \dfrac{i^{ε_2} }{\ln^β i · \ln^γ \ln i} ⇝ ∞⁺$ |
| $⇓$ | $α < 1 ⇒ \sum\limits_{i=2}^{∞⁺} s_i \not⇝ S$ | $⇐$ | $\int\limits_{i=2}^{∞⁺} \dfrac{1}{i^{1 - ε_1} } \mathrm{d}i \not⇝ I$ |
| $⇓$ | $α = 1 ⇒ \int\limits_{2}^{∞⁺} s_i \mathrm{d}i = \int\limits_{\ln 2}^{∞⁺} \dfrac{\mathrm{d}u}{u^β · \ln^γ u}$ | $⇐$ | $u ≡ \ln i ⇒ \dfrac{\mathrm{d}i}{i} = \mathrm{d}u$ |
| $⇓$ | $α = 1, β > 1 ⇒ \sum\limits_{i=2}^{∞⁺} s_i ⇝ S$ | $⇐$ | $β > 1 ⇒ \int\limits_{\ln 2}^{∞⁺} \dfrac{\mathrm{d}u}{u^β · \ln^γ u} ⇝ I$ |
| $⇓$ | $α = 1, β < 1 ⇒ \sum\limits_{i=2}^{∞⁺} s_i \not⇝ S$ | $⇐$ | $β < 1 ⇒ \int\limits_{\ln 2}^{∞⁺} \dfrac{\mathrm{d}u}{u^β · \ln^γ u} \not⇝ I$ |
附加证明：$\lim\limits_{i ⇝ ∞⁺} \dfrac{i^{ε_2} }{\ln^β i · \ln^γ \ln i} ⇝ ∞⁺$
$\lim\limits_{i⇝∞⁺} \dfrac{i^{ε_2} }{\ln^β i · \ln^γ \ln i} = \left[ \lim\limits_{i⇝∞⁺} \dfrac{i^{\frac{ε_2}{2·β} } }{\ln i} \right]^{β} · \left[ \lim\limits_{i⇝∞⁺} \dfrac{i^{\frac{ε_2}{2·γ} } }{\ln \ln i} \right]^{γ} = \left[ \lim\limits_{i⇝∞⁺} \dfrac{ε_2}{2·β} · i^{\frac{ε_2}{2·β} - 1 + 1} \right]^{β} · \left[ \lim\limits_{i⇝∞⁺} \dfrac{ε_2}{2·γ} · i^{\frac{ε_2}{2·γ}-1+1} · \ln i \right]^{γ} ⇝ ∞⁺$
正项级数的比值审敛法
| $⇑$ | $\left[ ∃N∈ℕ;∀i≥N; s_i ≤ \dfrac{\mathrm{Sup.} }{γ^i} \right] ⇒ \left[ γ > 1 ⇒ \sum\limits_{i=0}^{∞⁺} s_i ⇝ S \right]$ | $\left[ ∃N∈ℕ;∀i≥N; s_i ≥ \dfrac{\mathrm{Sup.} }{γ^i} \right] ⇒ \left[ 0 < γ < 1 ⇒ \sum\limits_{i=0}^{∞⁺} s_i \not⇝ S \right]$ | |
|---|---|---|---|
| $⇑$ | $∃N∈ℕ;∀i≥N; s_{i+1} · γ^{i + 1} ≤ s_i · γ^i ≤ \mathrm{Sup.}$ | $⇒$ | $\dfrac{s_i}{s_{i+1} } ≤ γ + o(1)$ |
| $⇑$ | $∃N∈ℕ;∀i≥N; \dfrac{s_i}{s_{i+1} } = γ_i + o (1)$ | $⇒$ | $γ_i = \dfrac{s_i}{s_{i+1} } + o (1)$ |
| $⇑$ | $\dfrac{s_i}{s_{i+1} } = γ_i ≥ γ > 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\dfrac{s_i}{s_{i+1} } = γ_i ≤ 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i \not⇝ S$ |
| $⇑$ | $\varliminf\limits_{i ⇝ ∞⁺} \dfrac{s_i}{s_{i+1} } = \varliminf\limits_{i ⇝ ∞⁺} γ_i ⇝ γ > 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\varlimsup\limits_{i ⇝ ∞⁺} \dfrac{s_i}{s_{i+1} } = \varlimsup\limits_{i ⇝ ∞⁺} γ_i ⇝ γ < 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i \not⇝ S$ |
特例：正项级数$\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} \dfrac{1}{i} \not⇝ S$，正项级数$\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} \dfrac{1}{i^2} ⇝ S$。
$\lim\limits_{i ⇝ ∞⁺} γ_i = \lim\limits_{i ⇝ ∞⁺} \dfrac{i + 1}{i} ⇝ 1$
$\lim\limits_{i ⇝ ∞⁺} γ_i = \lim\limits_{i ⇝ ∞⁺} \dfrac{i^2}{(i + 1)^2} = \left[ \lim\limits_{i ⇝ ∞⁺} \dfrac{i}{i + 1} \right]^2 ⇝ 1$
正项级数的根值审敛法
| $⇑$ | $\left[ ∃N∈ℕ;∀i≥N; s_i ≤ \dfrac{\mathrm{Sup.} }{γ^i} \right] ⇒ \left[ γ > 1 ⇒ \sum\limits_{i=0}^{∞⁺} s_i ⇝ S \right]$ | $\left[ ∃N∈ℕ;∀i≥N; s_i ≥ \dfrac{\mathrm{Sup.} }{γ^i} \right] ⇒ \left[ 0 < γ < 1 ⇒ \sum\limits_{i=0}^{∞⁺} s_i \not⇝ S \right]$ | |
|---|---|---|---|
| $⇑$ | $∃N∈ℕ;∀i≥N; \sqrt[i + 1]{s_{i+1} } · γ ≤ \sqrt[i]{s_i} · γ ≤ \sqrt[N]{\mathrm{Sup.} } ⇝ 1$ | $⇒$ | $\sqrt[i]{s_i} ≤ \dfrac{1}{γ} + o \left( \dfrac{1}{γ} \right)$ |
| $⇑$ | $∃N∈ℕ;∀i≥N; \sqrt[i]{s_i} = \dfrac{1}{γ_i} + o \left( \dfrac{1}{γ_i} \right)$ | $⇒$ | $γ_i = \dfrac{1}{\sqrt[i]{s_i} } + o (1)$ |
| $⇑$ | $\dfrac{1}{\sqrt[i]{s_i} } = γ_i ≥ γ > 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\dfrac{1}{\sqrt[i]{s_i} } = γ_i ≤ 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i \not⇝ S$ |
| $⇑$ | $\varliminf\limits_{i ⇝ ∞⁺} \dfrac{1}{\sqrt[i]{s_i} } = \varliminf\limits_{i ⇝ ∞⁺} γ_i ⇝ γ > 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\varlimsup\limits_{i ⇝ ∞⁺} \dfrac{1}{\sqrt[i]{s_i} } = \varlimsup\limits_{i ⇝ ∞⁺} γ_i ⇝ γ < 1$ | $⇒$ | $\sum\limits_{i=0}^{∞⁺} s_i \not⇝ S$ |
特例：正项级数$\sum\limits_{i = 1}^{∞⁺} \dfrac{1}{i} \not⇝ S$，正项级数$\sum\limits_{i = 1}^{∞⁺} \dfrac{1}{i^2} ⇝ S$。
$\lim\limits_{i ⇝ ∞⁺} γ_i = \lim\limits_{i ⇝ ∞⁺} \sqrt[i]{i} ⇝ 1$
$\lim\limits_{i ⇝ ∞⁺} γ_i = \lim\limits_{i ⇝ ∞⁺} \sqrt[i]{i^2} = \left[ \lim\limits_{i ⇝ ∞⁺} \sqrt[i]{i} \right]^2 ⇝ 1$
正项级数的幂值审敛法
| $⇑$ | $\left[ ∃N∈ℕ;∀i≥N; s_i ≤ \dfrac{\mathrm{Sup.} }{i^α} \right] ⇒ \left[ α > 1 ⇒ \sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i ⇝ S \right]$ | $\left[ ∃N∈ℕ;∀i≥N; s_i ≥ \dfrac{\mathrm{Sup.} }{i^α} \right] ⇒ \left[ α < 1 ⇒ \sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i \not⇝ S \right]$ | |
|---|---|---|---|
| $⇑$ | $∃N∈ℕ;∀i≥N; s_{i+1} · (i + 1)^α ≤ s_i · i^α ≤ \mathrm{Sup.}$ | $⇒$ | $\dfrac{s_i}{s_{i+1} } ≥ \dfrac{(i + 1)^α}{i^α} = 1 + \dfrac{α}{i} + o \left( \dfrac{1}{i} \right)$ |
| $⇑$ | $∃N∈ℕ;∀i≥N; \dfrac{s_i}{s_{i + 1} } = 1 + \dfrac{α_i}{i} + o \left( \dfrac{1}{i} \right)$ | $⇒$ | $α_i = i · \left( \dfrac{s_i}{s_{i+1} } - 1 \right) + o (1)$ |
| $⇑$ | $i · \left( \dfrac{s_i}{s_{i+1} } - 1 \right) = α_i ≥ α > 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $i · \left( \dfrac{s_i}{s_{i+1} } - 1 \right) = α_i ≤ 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i \not⇝ S$ |
| $⇑$ | $\varliminf\limits_{i ⇝ ∞⁺} i · \left( \dfrac{s_i}{s_{i+1} } - 1 \right) = \varliminf\limits_{i ⇝ ∞⁺} α_i ⇝ α > 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\varlimsup\limits_{i ⇝ ∞⁺} i · \left( \dfrac{s_i}{s_{i+1} } - 1 \right) = \varlimsup\limits_{i ⇝ ∞⁺} α_i ⇝ α < 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i \not⇝ S$ |
特例：正项级数$\sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} \dfrac{1}{i} \not⇝ S$，正项级数$\sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} \dfrac{1}{i · \ln^2 i} ⇝ S$。
$\lim\limits_{i ⇝ ∞⁺} α_i = \lim\limits_{i ⇝ ∞⁺} i · \left( \dfrac{i + 1}{i} - 1 \right) = \lim\limits_{i ⇝ ∞⁺} 1 ⇝ 1$
$\lim\limits_{i ⇝ ∞⁺} α_i = \lim\limits_{i ⇝ ∞⁺} i · \left[ \dfrac{(i + 1) · \ln^2 (i + 1)}{i · \ln^2 i} - 1 \right] = \lim\limits_{i ⇝ ∞⁺} \left[ (i + 1) · \left[ \lim\limits_{i ⇝ ∞⁺} \dfrac{\ln (i + 1)}{\ln i} \right]^2 - i \right] = \lim\limits_{i ⇝ ∞⁺} 1 ⇝ 1$
正项级数的指数审敛法
| $⇑$ | $\left[ ∃N∈ℕ;∀i≥N; s_i ≤ \dfrac{\mathrm{Sup.} }{i^α} \right] ⇒ \left[ α > 1 ⇒ \sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i ⇝ S \right]$ | $\left[ ∃N∈ℕ;∀i≥N; s_i ≥ \dfrac{\mathrm{Sup.} }{i^α} \right] ⇒ \left[ α < 1 ⇒ \sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i \not⇝ S \right]$ | |
|---|---|---|---|
| $⇑$ | $∃N∈ℕ;∀i≥N; s_{i+1} · (i + 1)^α ≤ s_i · i^α ≤ \mathrm{Sup.}$ | $⇒$ | $α ≤ \dfrac{\ln \mathrm{Sup.} - \ln s_i}{\ln i} = \dfrac{\ln s_i^{-1} }{\ln i} + o (1)$ |
| $⇑$ | $∃N∈ℕ;∀i≥N; α_i = \dfrac{\ln s_i^{-1} }{\ln i} + o (1)$ | $⇒$ | $s_i · i^{α + o (1)} ≤ 1$ |
| $⇑$ | $\dfrac{\ln s_i^{-1} }{\ln i} = α_i ≥ α > 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\dfrac{\ln s_i^{-1} }{\ln i} = α_i ≤ 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i \not⇝ S$ |
| $⇑$ | $\varliminf\limits_{i ⇝ ∞⁺} \dfrac{\ln s_i^{-1} }{\ln i} = \varliminf\limits_{i ⇝ ∞⁺} α_i ⇝ α > 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\varlimsup\limits_{i ⇝ ∞⁺} \dfrac{\ln s_i^{-1} }{\ln i} = \varlimsup\limits_{i ⇝ ∞⁺} α_i ⇝ α < 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i \not⇝ S$ |
| $⇑$ | $\varlimsup\limits_{i ⇝ ∞⁺} s_i · i^α ⇝ \mathrm{Sup.} < ∞⁺$ | $⇒$ | $α > 1 ⇒ \sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\varliminf\limits_{i ⇝ ∞⁺} s_i · i^α ⇝ \mathrm{Inf.} > 0$ | $⇒$ | $α ≤ 1 ⇒ \sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i \not⇝ S$ |
特例：正项级数$\sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} \dfrac{1}{i} \not⇝ S$，正项级数$\sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} \dfrac{1}{i · \ln^2 i} ⇝ S$。
$\lim\limits_{i ⇝ ∞⁺} α_i = \lim\limits_{i ⇝ ∞⁺} \dfrac{\ln i}{\ln i} = \lim\limits_{i ⇝ ∞⁺} 1 ⇝ 1$
$\lim\limits_{i ⇝ ∞⁺} α_i = \lim\limits_{i ⇝ ∞⁺} \dfrac{\ln \left( i · \ln^2 i \right)}{\ln i} = \lim\limits_{i ⇝ ∞⁺} \dfrac{\frac{1}{i · \ln i} · \left( \ln^2 i + i · 2 · \ln i · \frac{1}{i} \right)}{\frac{1}{i} } = \lim\limits_{i ⇝ ∞⁺} \left( 1 + \dfrac{2}{\ln i} \right) ⇝ 1$
$\lim\limits_{i ⇝ ∞⁺} s_i · i^1 = \lim\limits_{i ⇝ ∞⁺} \dfrac{1}{i} · i = \lim\limits_{i ⇝ ∞⁺} 1 ⇝ 1$
正项级数的对数审敛法
| $⇑$ | $\left[ ∃N∈ℕ;∀i≥N; s_i ≤ \dfrac{\mathrm{Sup.} }{i · \ln^β i} \right] ⇒ \left[ β > 1 ⇒ \sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} s_i ⇝ S \right]$ | $⇒$ | $\left[ ∃N∈ℕ;∀i≥N; s_i ≥ \dfrac{\mathrm{Sup.} }{i · \ln^β i} \right] ⇒ \left[ β < 1 ⇒ \sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} s_i \not⇝ S \right]$ |
|---|---|---|---|
| $⇑$ | $∃N∈ℕ;∀i>N; s_{i+1} · (i+1) · \ln^β (i+1) ≤ s_i · i · \ln^β i ≤ \mathrm{Sup.}$ | $⇒$ | $\dfrac{s_i}{s_{i+1} } ≥ \dfrac{i + 1}{i} · \dfrac{\ln^β (i + 1)}{\ln^β i} = 1 + \dfrac{1}{i} + \dfrac{β}{i · \ln i} + o \left( \dfrac{1}{i · \ln i} \right)$ |
| $⇑$ | $∃N∈ℕ;∀i>N; \dfrac{s_i}{s_{i+1} } = 1 + \dfrac{1}{i} + \dfrac{β_i}{i · \ln i} + o \left( \dfrac{1}{i · \ln i} \right)$ | $⇒$ | $β_i = i · \ln i · \left( \dfrac{s_i}{s_{i+1} } - \dfrac{i + 1}{i} \right) + o (1)$ |
| $⇑$ | $i · \ln i · \left( \dfrac{s_i}{s_{i+1} } - \dfrac{i + 1}{i} \right) = β_i ≥ β > 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $i · \ln i · \left( \dfrac{s_i}{s_{i+1} } - \dfrac{i + 1}{i} \right) = β_i ≤ 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} s_i \not⇝ S$ |
| $⇑$ | $\varliminf\limits_{i ⇝ ∞⁺} i · \ln i · \left( \dfrac{s_i}{s_{i+1} } - \dfrac{i + 1}{i} \right) = \varliminf\limits_{i ⇝ ∞⁺}β_i ⇝ β > 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} s_i ⇝ S$ |
| $⇑$ | $\varlimsup\limits_{i ⇝ ∞⁺} i · \ln i · \left( \dfrac{s_i}{s_{i+1} } - \dfrac{i + 1}{i} \right) = \varlimsup\limits_{i ⇝ ∞⁺} β_i ⇝ β < 1$ | $⇒$ | $\sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} s_i \not⇝ S$ |
附加证明：
$\dfrac{i + 1}{i} · \dfrac{\ln^β (i + 1)}{\ln^β i} = \left( 1 + \dfrac{1}{i} \right) · \left[ \dfrac{\ln i + \ln \left( 1 + \dfrac{1}{i} \right)}{\ln i} \right]^β = \left( 1 + \dfrac{1}{i} \right) · \left[ 1 + \dfrac{1}{i · \ln i} + o \left( \dfrac{1}{i · \ln i} \right) \right]^β = \left( 1 + \dfrac{1}{i} \right) · \left[ 1 + \dfrac{β}{i · \ln i} + o \left( \dfrac{1}{i · \ln i} \right) \right] = 1 + \dfrac{1}{i} + \dfrac{β}{i · \ln i} + o \left( \dfrac{1}{i · \ln i} \right)$
特例：正项级数$\sum\limits_{i=\rlap{≡≡}{0,1,}2}^{∞⁺} \dfrac{1}{i · \ln i} \not⇝ S$，正项级数$\sum\limits_{i=\rlap{≡≡}{0,1,}{2} }^{∞⁺} \dfrac{1}{i · \ln i · \ln^2 \ln i} ⇝ S$。
$\lim\limits_{i ⇝ ∞⁺} β_i = \lim\limits_{i ⇝ ∞⁺} i · \ln i · \left[ \dfrac{(i + 1) · \ln (i + 1)}{i · \ln i} - \dfrac{i + 1}{i} \right] = \lim\limits_{i ⇝ ∞⁺} \left[ (i + 1) · \ln (i + 1) - (i + 1) · \ln i \right] = \lim\limits_{i ⇝ ∞⁺} \left[ \ln \left( 1 + \dfrac{1}{i} \right)^i + \ln \left( 1 + \dfrac{1}{i} \right) \right] ⇝ 1$
$\lim\limits_{i ⇝ ∞⁺} β_i = \lim\limits_{i ⇝ ∞⁺} i · \ln i · \left[ \dfrac{(i + 1) · \ln (i + 1) · \ln^2 \ln (i + 1)}{i · \ln i · \ln^2 \ln i} - \dfrac{i + 1}{i} \right] = \lim\limits_{i ⇝ ∞⁺} \left[ (i + 1) · \ln (i + 1) · \left[ \lim\limits_{i ⇝ ∞⁺} \dfrac{\ln \ln (i + 1)}{\ln \ln i} \right]^2 - (i + 1) · \ln i \right] = \lim\limits_{i ⇝ ∞⁺} \left[ \ln \left( 1 + \dfrac{1}{i} \right)^i + \ln \left( 1 + \dfrac{1}{i} \right) \right] ⇝ 1$
正项级数的审敛法对比
收敛速度：比值审敛法 > 根值审敛法 > 幂值审敛法 > 指数审敛法 > 对数审敛法 > 比较审敛法
强效程度：比值审敛法 < 根值审敛法 < 幂值审敛法 < 指数审敛法 < 对数审敛法 < 比较审敛法
由于$\left[ 1 < \varliminf\limits_{i ⇝ ∞⁺} \dfrac{s_i}{s_{i+1} } ≤ \varliminf\limits_{i ⇝ ∞⁺} {\dfrac{1}{\sqrt[i]{s_i} } } \right] ≤ \varlimsup\limits_{i ⇝ ∞⁺} \dfrac{1}{\sqrt[i]{s_i} } ≤ \varlimsup\limits_{i ⇝ ∞⁺} \dfrac{s_i}{s_{i+1} }$
也即$\varliminf\limits_{i ⇝ ∞⁺} \dfrac{s_{i+1} }{s_i} ≤ \varliminf\limits_{i ⇝ ∞⁺} \sqrt[i]{s_i} ≤ \left[ \varlimsup\limits_{i ⇝ ∞⁺} \sqrt[i]{s_i} ≤ \varlimsup\limits_{i ⇝ ∞⁺} \dfrac{s_{i+1} }{s_i} < 1 \right]$
| $⇓$ | $\varlimsup\limits_{i ⇝ ∞⁺} \dfrac{s_{i+1} }{s_i} ⇝ ρ$ | $⇒$ | $∀ε>0;∃N∈ℕ;∀n≥N; \dfrac{s_{i+1} }{s_i} < ρ + ε$ |
|---|---|---|---|
| $⇓$ | $\dfrac{s_i}{s_N} = \dfrac{s_i}{s_{i-1} } ··· \dfrac{s_{N-1} }{s_N} < (ρ + ε)^{i - N}$ | $⇒$ | $s_i < s_N · (ρ + ε)^{-N} · (ρ + ε)^i$ |
| $⇓$ | $\sqrt[i]{s_i} < \sqrt[i]{s_N · (ρ + ε)^{-N} } · (ρ + ε)$ | $⇒$ | $\varlimsup\limits_{i ⇝ ∞⁺} \sqrt[i]{s_i} ≤ \varlimsup\limits_{i ⇝ ∞⁺} \left[ \sqrt[i]{s_N · (ρ + ε)^{-N} } · (ρ + ε) \right] ⇝ ρ + ε$ |
| $⇓$ | $ε ⇝ 0⁺$ | $⇒$ | $\varlimsup\limits_{i ⇝ ∞⁺} \sqrt[i]{s_i} ≤ \varlimsup\limits_{i ⇝ ∞⁺} \dfrac{s_{i+1} }{s_i} ⇝ ρ$ |
| $⇓$ | $\varliminf\limits_{i ⇝ ∞⁺} \dfrac{s_i}{s_{i+1} } ≤ \varliminf\limits_{i ⇝ ∞⁺} {\dfrac{1}{\sqrt[i]{s_i} } }$ |
特例：正项级数$\sum\limits_{i=0 \ j=2·i,2·i+1}^{∞⁺} \left( \dfrac{1}{2^{2 · i} } + \dfrac{1}{3^{2 · i + 1} } \right) = \sum\limits_{i=0}^{∞⁺} \dfrac{1}{2^{2 · i} } + \dfrac{1}{3} · \sum\limits_{i=0}^{∞⁺} \dfrac{1}{3^{2 · i} } = \dfrac{1}{1 - \frac{1}{2^2} } + \dfrac{1}{3} · \dfrac{1}{1 - \frac{1}{3^2} } = \dfrac{41}{24}$
根据比值审敛法不可判定敛散性，但根据根值审敛法可判定为收敛。
$\varliminf\limits_{j⇝∞⁺} \dfrac{s_j}{s_{j+1} } = \min\left\lbrace \lim\limits_{i⇝∞⁺} \dfrac{1/2^{2·i} }{1/3^{2·i+1} }, \lim\limits_{i⇝∞⁺} \dfrac{1/3^{2·i+1} }{1/2^{2·i+2} } \right\rbrace = \lim\limits_{i⇝∞⁺} \dfrac{2^{2·i+2} } {3^{2·i+1} } ⇝ 0 < 1$
$\varlimsup\limits_{j⇝∞⁺} \dfrac{s_j}{s_{j+1} } = \max\left\lbrace \lim\limits_{i⇝∞⁺} \dfrac{1/2^{2·i} }{1/3^{2·i+1} }, \lim\limits_{i⇝∞⁺} \dfrac{1/3^{2·i+1} }{1/2^{2·i+2} } \right\rbrace = \lim\limits_{i⇝∞⁺} \dfrac{3^{2·i+1} }{2^{2·i} } ⇝ ∞⁺ > 1$
$\varliminf\limits_{j⇝∞⁺} \dfrac{1}{\sqrt[j]{s_j} } = \min\left\lbrace \lim\limits_{i⇝∞⁺} \sqrt[2·i]{2^{2·i} }, \lim\limits_{i⇝∞⁺} \sqrt[2·i+1]{3^{2·i+1} } \right\rbrace = \lim\limits_{i⇝∞⁺} \sqrt[2·i]{2^{2·i} } = 2 > 1$
$\varliminf\limits_{j⇝∞⁺} \dfrac{1}{\sqrt[j]{s_j} } = \max\left\lbrace \lim\limits_{i⇝∞⁺} \sqrt[2·i]{2^{2·i} }, \lim\limits_{i⇝∞⁺} \sqrt[2·i+1]{3^{2·i+1} } \right\rbrace = \lim\limits_{i⇝∞⁺} \sqrt[2·i+1]{3^{2·i+1} } = 3 > 1$
构造性收敛正项级数
不存在收敛最慢的正项级数。
假设存在正项级数$\sum\limits_{i=0}^{∞⁺} s_i ⇝ S$，则可构造收敛更慢的正项级数$t_i ≡ \sqrt{S - S_{i-1} } - \sqrt{S - S_i}$，使得$\sum\limits_{i=0}^{∞⁺} t_i ⇝ \sqrt{S}$。
| $⇓$ | $\sum\limits_{i=0}^{∞⁺} s_i ⇝ S$ | $\mathop{⇒}\limits^{S_{-1}≡0}$ | $s_i = S_i - S_{i-1} = (S - S_{i-1}) - (S - S_i)$ |
|---|---|---|---|
| $⇓$ | $1 = \dfrac{s_i}{(S - S_{i-1}) - (S - S_i)} = \dfrac{s_i}{(\sqrt{S - S_{i-1} } + \sqrt{S - S_i}) · (\sqrt{S - S_{i-1} } - \sqrt{S - S_i})}$ | $⇒$ | $t_i ≡ \sqrt{S - S_{i-1} } - \sqrt{S - S_i}, \sum\limits_{i=1}^{∞⁺} t_i ⇝ \sqrt{S - S_{-1} } = \sqrt{S}$ |
| $⇓$ | $\dfrac{s_i}{t_i} = \sqrt{S - S_{i-1} } + \sqrt{S - S_i}$ | $⇒$ | $\lim\limits_{i ⇝ ∞⁺} \dfrac{s_i}{t_i} ⇝ 0$ |
无穷级数的乘积
级数的乘积可类比于矩阵的乘法运算。
$\sum\limits_{i=0}^n s_i · \sum\limits_{j=0}^m t_j =
\sum\limits_{i=0,j=0}^{n,m}\left[\begin{matrix}
s_0 \ s_1 \ \vdots \ s_n
\end{matrix}\right] ·
\left[\begin{matrix}
t_0 & t_1 & \cdots & t_m
\end{matrix}\right] =
\sum\limits_{i=0,j=0}^{n,m}\left[\begin{matrix}
s_0 · t_0 & s_0 · t_1 & \cdots & s_0 · t_m
s_1 · t_0 & s_1 · t_1 & \cdots & s_1 · t_m
\vdots & \vdots & \ddots & \vdots
s_n · t_0 & s_n · t_1 & \cdots & s_n · t_m
\end{matrix}\right]$
级数的乘积沿左上正方形对角线相加。
$\sum\limits_{i=0}^{∞⁺} s_i · \sum\limits_{j=0}^{∞⁺} t_j = \sum\limits_{l=0}^{∞⁺} \left[ \sum\limits_{k=0}^l s_k · t_{l-k} \right] = \sum\limits_{l=0}^{∞⁺} \left[ \sum\limits_{i+j=l} s_i · t_j \right] = (s_0 · t_0) + (s_0 · t_1 + s_1 · t_0) + \cdots + (s_0 · t_l + s_1 · t_{l-1} + \cdots + s_{l-1} · t_1 + s_l · t_0) + \cdots$
级数的乘积按左上三角形体横向相加。
$\sum\limits_{i=0}^{∞⁺} s_i · \sum\limits_{j=0}^{∞⁺} t_j = \lim\limits_{l⇝∞⁺} \sum\limits_{i=0}^l \left[ s_i · \sum\limits_{j=0}^{l-i} t_j \right] = \lim\limits_{l⇝∞⁺} \sum\limits_{i=0}^l s_i · T_{l-i} = \lim\limits_{l⇝∞⁺} [ s_0 · (t_0 + t_1 + ··· + t_l) + s_1 · (t_0 + t_1 + ··· + t_{l-1}) + ··· + s_{l-1} · (t_0 + t_1) + s_l · (t_0) ]$
级数的乘积按左上三角形体竖向相加。
$\sum\limits_{i=0}^{∞⁺} s_i · \sum\limits_{j=0}^{∞⁺} t_j = \lim\limits_{l⇝∞⁺} \sum\limits_{j=0}^l \left[ t_j · \sum\limits_{i=0}^{l-j} s_i \right] = \lim\limits_{l⇝∞⁺} \sum\limits_{j=0}^l t_j · S_{l-j} = \lim\limits_{l⇝∞⁺} [ t_0 · (s_0 + s_1 + ··· s_l) + t_1 · (s_0 + s_1 + ··· + s_{l-1}) + ··· + t_{l-1} · (s_0 + s_1) + t_l · (s_0) ]$
级数的乘积沿左上正方形外边缘相加。
$\sum\limits_{i=0}^{∞⁺} s_i · \sum\limits_{j=0}^{∞⁺} t_j = \sum\limits_{l=0}^{∞⁺} \left[ - s_l · t_l + \sum\limits_{k=0}^l (s_k · t_l + s_l · t_k) \right] = (s_0 · t_0) + (s_0 · t_1 + s_1 · t_1 + s_1 · t_0) + \cdots + (s_0 · t_l + s_1 · t_l + \cdots + s_l · t_l + \cdots + s_l · t_1 + s_l · t_0) + \cdots$
级数的乘积按左上正方形体横向相加。
$\sum\limits_{i=0}^{∞⁺} s_i · \sum\limits_{j=0}^{∞⁺} t_j = \lim\limits_{l⇝∞⁺} \sum\limits_{i=0}^l \left[ s_i · \sum\limits_{j=0}^l t_j \right] = \lim\limits_{l⇝∞⁺} \sum\limits_{i=0}^l s_i · T_l = \lim\limits_{l⇝∞⁺} \left[ s_0 · (t_0 + t_1 + ··· + t_l) + s_1 · (t_0 + t_1 + ··· + t_l) + ··· + s_{l-1} · (t_0 + t_1 + ··· + t_l) + s_l · (t_0 + t_1 + ··· + t_l) \right]$
级数的乘积按左上正方形体竖向相加。
$\sum\limits_{i=0}^{∞⁺} s_i · \sum\limits_{j=0}^{∞⁺} t_j = \lim\limits_{l⇝∞⁺} \sum\limits_{j=0}^l \left[ t_j · \sum\limits_{i=0}^l s_i \right] = \lim\limits_{l⇝∞⁺} \sum\limits_{j=0}^l t_j · S_l = \lim\limits_{l⇝∞⁺} [ t_0 · (s_0 + s_1 + ··· + s_l) + t_1 · (s_0 + s_1 + ··· + s_l) + ··· + t_{l-1} · (s_0 + s_1 + ··· + s_l) + t_l · (s_0 + s_1 + ··· + s_l) ]$
无穷级数的乘积收敛
若无穷级数之一绝对收敛，且其余无穷级数条件收敛，则无穷级数的乘积必定绝对收敛。
| $\left[ \sum\limits_{i=0}^{∞⁺} | s_i | ⇝ S_{ | } \right] ∧ \left[ \sum\limits_{j=0}^{∞⁺} t_j ⇝ T \right] ⇒ \left[ \lim\limits_{l⇝∞⁺} \sum\limits_{i=0}^l \left | s_i · \sum\limits_{j=0}^l t_j \right | ⇝ ST_{ | } \right]$ |
| $⇓$ | $\sum\limits_{i=1}^{∞⁺} s_i · \sum\limits_{j=1}^{∞⁺} t_j = \lim\limits_{l⇝∞⁺} \sum\limits_{i=1}^l \left[ s_i · \sum\limits_{j=1}^l t_j \right] = \lim\limits_{l⇝∞⁺} \sum\limits_{i=1}^l s_i · T_l$ | $⇒$ | $\lim\limits_{l⇝∞⁺} \sum\limits_{i=1}^l \left | s_i · \sum\limits_{j=1}^l t_j \right | = \lim\limits_{l⇝∞⁺} \sum\limits_{i=1}^l | s_i · T_l | $ | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $⇓$ | $\lim\limits_{l⇝∞⁺} \sum\limits_{i=1}^l | s_i · T_l | = \lim\limits_{l⇝∞⁺} \left[ | T | · \sum\limits_{i=1}^l | s_i | + \sum\limits_{i=1}^l | s_i | · ( | T_l | - | T | ) \right]$ | ||||||||||||
| $⇓$ | $\lim\limits_{l⇝∞⁺} \sum\limits_{i=1}^l | s_i · T_l | ≤ \lim\limits_{l⇝∞⁺} \left[ | T | · \sum\limits_{i=1}^l | s_i | + \sum\limits_{i=1}^L | s_i | · ( | T_l | - | T | ) + \sum\limits_{i=L+1}^l | s_i | · ( | T_l | - | T | ) \right]$ | ||||||
| $⇓$ | $\lim\limits_{l⇝∞⁺} \sum\limits_{i=1}^l | s_i · T_l | ≤ | T | · \sum\limits_{i=1}^{∞⁺} | s_i | + ε_1 · \sum\limits_{i=1}^L | s_i | + \mathrm{Sup.} · \sum\limits_{i=L+1}^l | s_i | $ | $⇐$ | $∃L∈ℕ;[ l>L ⇒ | T_l | - | T | < ε_1 ] ∧ [ l≤L ⇒ | T_l | - | T | ≤ \mathrm{Sup.}]$ | ||||
| $⇓$ | $\lim\limits_{l⇝∞⁺} \sum\limits_{i=1}^l | s_i · T_l | ≤ | T | · \sum\limits_{i=1}^{∞⁺} | s_i | + ε_1 · S_{ | } + \mathrm{Sup.} · ε_2$ | |||||||||||||||||
| $⇓$ | $\lim\limits_{l⇝∞⁺} \sum\limits_{i=1}^l | s_i · T_l | ≤ | T | · S_{ | } + ε$ | $⇐$ | $ε = ε_1 · S_{ | } + \mathrm{Sup.} · ε_2$ | ||||||||||||||||
| $⇓$ | $\lim\limits_{l⇝∞⁺} \sum\limits_{i=1}^l \left | s_i · \sum\limits_{j=1}^l t_j \right | ⇝ ST_{ | }$ |
若无穷级数全都条件收敛，则无穷级数的乘积未必条件收敛。
若无穷级数全都条件收敛，且无穷级数的乘积条件收敛，则该乘积等于无穷级数的极限相乘。
$\left[ \sum\limits_{i=0}^{∞⁺} s_i = S \right] ∧ \left[ \sum\limits_{j=0}^{∞⁺} t_j = T \right] ∧ \left[ \sum\limits_{l=0}^{∞⁺} \left( \sum\limits_{k=0}^l s_k · t_{l-k} \right) = ST \right] ⇒ \left[ S · T = \sum\limits_{i=0}^{∞⁺} s_i · \sum\limits_{j=0}^{∞⁺} t_j = \sum\limits_{l=0}^{∞⁺} \left( \sum\limits_{k=0}^l s_k · t_{l-k} \right) = ST \right]$
典例：无穷级数$\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i = \sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} \dfrac{(-1)^{i} }{\sqrt{i} } ⇝ S$条件收敛，无穷级数$\sum\limits_{j=\rlap{≡}{0,}1}^{∞⁺} t_i = \sum\limits_{j=\rlap{≡}{0,}1}^{∞⁺} \dfrac{(-1)^j}{\sqrt{j} } ⇝ T$条件收敛, 两者的乘积沿对角线相加$\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} s_i · \sum\limits_{j=\rlap{≡}{0,}1}^{∞⁺} t_j \not⇝ ST$。
| $\lim\limits_{l⇝∞⁺} \left | \sum\limits_{k=1}^l s_k · t_{l-k} \right | = \lim\limits_{l⇝∞⁺} \left | \sum\limits_{k=1}^l \dfrac{(-1)^k}{\sqrt{k} } · \dfrac{(-1)^{l - k} }{\sqrt{l - k} } \right | = \lim\limits_{l⇝∞⁺} \sum\limits_{k=1}^l \dfrac{1}{\sqrt{k · (l - k)} } ≥ \lim\limits_{l⇝∞⁺} \dfrac{l}{\frac{k + (l - k)}{2} } ⇝ 2 ≠ 0$ |
无穷乘积
无穷乘积的部分积
$P_l ≡ \prod\limits_{n=0}^l p_n ≡ \prod\limits_{n=0}^l (1 + s_n)$
无穷乘积的极限
$\lim\limits_{l⇝∞⁺} P_l ≡ \lim\limits_{l⇝∞⁺} \prod\limits_{n=0}^l p_n ≡ \prod\limits_{n=0}^{∞⁺} p_n ≡ \prod\limits_{n=0}^{∞⁺} (1 + s_n)$
无穷乘积收敛
| $\lim\limits_{l⇝∞⁺} P_l ⇝ P ≠ 0$，也即$\lim\limits_{l⇝∞⁺} | P_l - P | ⇝ 0$ |
| $\prod\limits_{n=0}^{∞⁺} p_n ⇝ P ≠ 0$，也即$\lim\limits_{n⇝∞⁺} \left | \prod\limits_{n=0}^n p_n - P \right | ⇝ 0$ |
无穷乘积发散
| $\lim\limits_{l⇝∞⁺} P_l \not⇝ P ≠ 0$，也即$\lim\limits_{l⇝∞⁺} | P_l - P | \not⇝ 0$ |
| $\prod\limits_{n=0}^{∞⁺} p_n \not⇝ P ≠ 0$，也即$\lim\limits_{n⇝∞⁺} \left | \prod\limits_{n=0}^n p_n - P \right | \not⇝ 0$ |
无穷乘积收敛的性质
若无穷乘积收敛，则其通项的极限为一。
$\lim\limits_{l⇝∞⁺} P_l ⇝ P ≠ 0 ⇒ \lim\limits_{n⇝∞⁺} p_n ⇝ 1 ⇒ \lim\limits_{n⇝∞⁺} s_n ⇝ 0$
$\lim\limits_{n⇝∞⁺} p_n = \lim\limits_{n⇝∞⁺} \dfrac{P_n}{P_{n-1} } = \dfrac{\lim\limits_{n⇝∞⁺} P_n}{\lim\limits_{n⇝∞⁺} P_{n-1} } = \dfrac{P}{P} ⇝ 1$
若通项的极限不为一，则无穷乘积发散。
$\lim\limits_{n⇝∞⁺} p_n \not⇝ 1 ⇒ \lim\limits_{l⇝∞⁺} P_l \not⇝ P ≠ 0$
无穷乘积的对偶原理
若无穷乘积收敛，则其对应的无穷级数也收敛，反之亦然。
$\left[ \prod\limits_{n=0}^{∞⁺} p_n ⇝ P \right] ⇔ \left[ \ln \prod\limits_{n=0}^{∞⁺} p_n ⇝ \ln P \right] ⇔ \left[ \sum\limits_{n=0}^{∞⁺} \ln p_n ⇝ \ln P \right]$
| $\left[ \prod\limits_{n=0}^{∞⁺} (1 + | s_n | ) ⇝ P_{ | } \right] ⇔ \left[ \sum\limits_{n=0}^{∞⁺} \ln (1 + | s_n | ) ⇝ \ln P_{ | } \right] ⇔ \left[ \sum\limits_{n=0}^{∞⁺} | s_n | ⇝ S_{ | } \right]$ |
$\left[ \prod\limits_{n=0}^{∞⁺} (1 + s_n) ⇝ P \right] ⇔ \left[ \sum\limits_{n=0}^{∞⁺} \ln (1 + s_n) ⇝ \ln P \right] \mathop{⇒}\limits_{-1 < s_n ≤ 0} \left[ \sum\limits_{n=0}^{∞⁺} s_n ⇝ S \right]$
若无穷乘积发散于零，则其对应的无穷级数发散于负无穷。
$\left[ \prod\limits_{n=0}^{∞⁺} p_n ⇝ 0 \right] ⇔ \left[ \sum\limits_{n=0}^{∞⁺} \ln p_n ⇝ ∞⁻ \right]$
$\left[ \prod\limits_{n=0}^{∞⁺} (1 + s_n) ⇝ 0 \right] ⇔ \left[ \sum\limits_{n=0}^{∞⁺} \ln (1 + s_n) ⇝ ∞⁻ \right] \mathop{⇐}\limits_{-1 < s_n ≤ 0} \left[ \sum\limits_{n=0}^{∞⁺} s_n ⇝ ∞⁻ \right]$
| $⇓$ | $(1 + | x | ) · (1 + | y | ) = 1 + | x | + | y | + | x | · | y | ≥ 1 + | x | + | y | > | x | + | y | $ | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $⇓$ | $\prod\limits_{n=0}^{∞⁺} (1 + | s_n | ) > \sum\limits_{n=0}^{∞⁺} | s_n | ≥ \sum\limits_{n=0}^{∞⁺} \ln (1 + | s_n | ) ≥ 0$ | $⇐$ | $ | s_n | ≥ \ln (1 + | s_n | ) ≥ 0$ | ||||||||||
| $⇓$ | $\prod\limits_{n=0}^{∞⁺} (1 + | s_n | ) ⇝ P_{ | } ⇒ \sum\limits_{n=0}^{∞⁺} | s_n | ⇝ S_{ | } ⇒ \sum\limits_{n=0}^{∞⁺} \ln (1 + | s_n | ) ⇝ \ln P_{ | }$ | |||||||||||||
| $⇓$ | $\prod\limits_{n=0}^{∞⁺} (1 + | s_n | ) ⇝ P_{ | } ⇔ \sum\limits_{n=0}^{∞⁺} | s_n | ⇝ S_{ | } ⇔ \sum\limits_{n=0}^{∞⁺} \ln (1 + | s_n | ) ⇝ \ln P_{ | }$ | |||||||||||||
| $⇓$ | $-1 < s_n ≤ 0 ⇒ \prod\limits_{n=0}^{∞⁺} (1 + s_n) > 0 ≥ \sum\limits_{n=0}^{∞⁺} s_n ≥ \sum\limits_{n=0}^{∞⁺} \ln (1 + s_n)$ | $⇐$ | $ | \ln (1 + s_n) | ≥ | s_n | ≥ \ln (1 + | s_n | ) ≥ s_n ≥ \ln (1 + s_n) \mathop{=}\limits_{-1 < s_n} \ln | 1 + s_n | $ | ||||||||||||
| $⇓$ | $\prod\limits_{n=0}^{∞⁺} (1 + s_n) ⇝ P ⇔ \sum\limits_{n=0}^{∞⁺} \ln (1 + s_n) ⇝ \ln P ⇒ \sum\limits_{n=0}^{∞⁺} s_n ⇝ S$ |
特例：若$\sum\limits_{n=0}^{∞⁺} s_n^2 ⇝ \mathrm{Con.}$，则$\left[ \sum\limits_{n=0}^{∞⁺} s_n ⇝ S \right] ⇔ \left[ \sum\limits_{n=0}^{∞⁺} \ln (1 + s_n) ⇝ \ln P \right] ⇔ \left[ \prod\limits_{n=0}^{∞⁺} (1 + s_n) ⇝ P \right]$
| $⇓$ | $\sum\limits_{n=1}^{∞⁺} s_n^2 ⇝ \mathrm{Con.}$ |
|---|---|
| $⇓$ | $\lim\limits_{n⇝∞⁺} s_n^2 ⇝ 0 ⇒ \lim\limits_{n⇝∞⁺} s_n ⇝ 0 ⇒ \lim\limits_{n⇝∞⁺} \ln (1 + s_n) ⇝ 0$ |
| $⇓$ | $\lim\limits_{n⇝∞⁺} \dfrac{s_n - \ln (1 + s_n)}{s_n^2} = \lim\limits_{n⇝∞⁺} \dfrac{1}{2 · (1 + s_n)} ⇝ \dfrac{1}{2}$ |
| $⇓$ | $\sum\limits_{n=1}^{∞⁺} s_n^2 ⇝ \mathrm{Con.} ⇒ \left[ \sum\limits_{n=1}^{∞⁺} s_n - \sum\limits_{n=1}^{∞⁺} \ln (1 + s_n) ⇝ \mathrm{Con.} \right]$ |
| $⇓$ | $\sum\limits_{n=1}^{∞⁺} s_n^2 ⇝ \mathrm{Con.} ⇒ \left[ \sum\limits_{n=1}^{∞⁺} s_n ⇝ S ⇔ \sum\limits_{n=1}^{∞⁺} \ln (1 + s_n) ⇝ \ln P ⇔ \prod\limits_{n=1}^{∞⁺} (1 + s_n) ⇝ P \right]$ |
| $⇓$ | $\sum\limits_{n=1}^{∞⁺} s_n^2 ⇝ ∞⁺ ⇒ \left[ \sum\limits_{n=1}^{∞⁺} s_n - \sum\limits_{n=1}^{∞⁺} \ln (1 + s_n) ⇝ ∞⁺ \right]$ |
| $⇓$ | $\sum\limits_{n=1}^{∞⁺} s_n^2 ⇝ ∞⁺ ⇒ \left[ \sum\limits_{n=1}^{∞⁺} s_n ⇝ S ⇔ \sum\limits_{n=1}^{∞⁺} \ln (1 + s_n) ⇝ ∞⁻ ⇔ \prod\limits_{n=1}^{∞⁺} (1 + s_n) ⇝ 0 \right]$ |
无穷乘积的绝对收敛与条件收敛
若无穷乘积绝对收敛，则必定条件收敛。
| $\left[ \prod\limits_{n=0}^{∞⁺} (1 + | s_n | ) ⇝ P_{ | } \right] ⇔ \left[ \sum\limits_{n=0}^{∞⁺} \ln (1 + | s_n | ) ⇝ \ln P_{ | } \right] ⇒ \left[ \sum\limits_{n=0}^{∞⁺} \ln (1 + s_n) ⇝ \ln P \right] ⇔ \left[ \prod\limits_{n=0}^{∞⁺} (1 + s_n) ⇝ P \right]$ |
若无穷乘积条件收敛，则未必绝对收敛。
| $\left[ \prod\limits_{n=0}^{∞⁺} (1 + s_n) ⇝ P \right] ⇔ \left[ \sum\limits_{n=0}^{∞⁺} \ln (1 + s_n) ⇝ \ln P \right] \not⇒ \left[ \sum\limits_{n=0}^{∞⁺} \ln (1 + | s_n | ) ⇝ \ln P_{ | } \right] ⇔ \left[ \prod\limits_{n=0}^{∞⁺} (1 + | s_n | ) ⇝ P_{ | } \right]$ |
无穷乘积绝对收敛与条件收敛的性质
若无穷乘积绝对收敛，则交换任意多项的次序，其极限不变并且保持绝对收敛。
若无穷乘积条件收敛，则交换任意多项的次序，可使其趋近于任意给定的正数。
函数项级数
函数项级数的部分和
$F_n (x) ≡ \sum\limits_{i=0}^n f_i (x)$
函数项级数的极限
$\lim\limits_{n⇝∞⁺} F_n (x) ≡ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n f_i (x) ≡ \sum\limits_{i=0}^{∞⁺} f_i (x)$
函数项级数收敛
| $\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} F_n (x) ⇝ F(x_0)$也即$\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} | F_n (x) - F (x_0) | ⇝ 0$ |
| $\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n f_i (x) ⇝ F (x_0)$也即$\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} \left | \sum\limits_{i=0}^n f_i (x) - F (x_0) \right | ⇝ 0$ |
函数项级数发散
| $\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} F_n (x) \not⇝ F(x_0)$也即$\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} | F_n (x) - F (x_0) | \not⇝ 0$ |
| $\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n f_i (x) \not⇝ F (x_0)$也即$\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} \left | \sum\limits_{i=0}^n f_i (x) - F (x_0) \right | \not⇝ 0$ |
典例：函数项级数$F_n (x) = x^0 + \sum\limits_{i=1}^n (x^i - x^{i-1}) = x^n$
$F (x) = \lim\limits_{n⇝∞⁺} F_n (x) = \lim\limits_{n⇝∞⁺} x^n$
函数项级数$F_n (x)$在收敛区间$(-1, 1]$内都连续且可导，在点$x = 1$处不连续也不可导。
| $x < -1$ | $⇒$ | $\lim\limits_{n⇝∞⁺} x^n ⇝ ∞$ | ||
|---|---|---|---|---|
| $x = -1$ | $⇒$ | $\lim\limits_{n⇝∞⁺} x^n \not⇝ \mathrm{Lim.}$ | ||
| $ | x | < +1$ | $⇒$ | $\lim\limits_{n⇝∞⁺} x^n ⇝ 0$ |
| $x = +1$ | $⇒$ | $\lim\limits_{n⇝∞⁺} x^n ⇝ 1$ | ||
| $x > +1$ | $⇒$ | $\lim\limits_{n⇝∞⁺} x^n ⇝ ∞⁺$ |
函数项级数的一致收敛
函数项级数一致收敛
| $\left[ ∀ε>0;∃N∈ℕ;∀n≥N; \sup\limits_{x∈X} | F_n (x) - F (x) | ≤ ε \right] ⇔ [∀ε>0;∃N∈ℕ;∀n≥N;∀x∈X; | F_n (x) - F (x) | < ε]$ |
| $\left[ ∀ε>0;∃N∈ℕ;∀n≥N;∀m>N; \sup\limits_{x∈X} | F_m (x) - F_n (x) | ≤ ε \right] ⇔ [∀ε>0;∃N∈ℕ;∀n≥N;∀m>N;∀x∈X; | F_m (x) - F_n (x) | < ε]$ |
| $\left[ ∀ε>0;∃N∈ℕ;∀n≥N;∀m>n; \sup\limits_{x∈X} \left | \sum\limits_{i=n}^m f_i (x) \right | ≤ ε \right] ⇔ \left[ ∀ε>0;∃N∈ℕ;∀n≥N;∀m>n;∀x∈X; \left | \sum\limits_{i=n}^m f_i (x) \right | < ε \right]$ |
| $\left[ \lim\limits_{n⇝∞⁺} F_n (x) \mathop{↭}\limits_{x∈X} F (x) \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} | F_n (x) - F (x) | \mathop{↭}\limits_{x∈X} 0 \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} \sup\limits_{x∈X} | F_n (x) - F (x) | ⇝ 0 \right]$ |
| $\left[ \sum\limits_{i=0}^{∞⁺} f_i (x) \mathop{↭}\limits_{x∈X} F (x) \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} \left | \sum\limits_{i=0}^n f_i (x) - F (x) \right | \mathop{↭}\limits_{x∈X} 0 \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} \sup\limits_{x∈X} \left | \sum\limits_{i=0}^n f_i (x) - F (x) \right | ⇝ 0 \right]$ |
若函数项级数一致收敛，则其通项一致收敛于零。
| $\lim\limits_{n⇝∞⁺} F_n (x) \mathop{↭}\limits_{x∈X} F (x) ⇒ \lim\limits_{i⇝∞⁺} f_i (x) \mathop{↭}\limits_{x∈X} 0 ⇔ \lim\limits_{i⇝∞⁺} \sup\limits_{x∈X} | f_i (x) | ⇝ 0 $ |
函数项级数非一致收敛
| $[∃ε>0;∀N∈ℕ;∃n≥N; \sup\limits_{x∈X} | F_n (x) - F (x) | > ε] ⇔ [∃ε>0;∀N∈ℕ;∃n≥N;∃x∈X; | F_n (x) - F (x) | ≥ ε]$ |
| $[∃ε>0;∀N∈ℕ;∃n≥N;∃m>N; \sup\limits_{x∈X} | F_m (x) - F_n (x) | > ε] ⇔ [∃ε>0;∀N∈ℕ;∃n≥N;∃m>N;∃x∈X; | F_m (x) - F_n (x) | ≥ ε]$ |
| $[∃ε>0;∀N∈ℕ;∃n≥N;∃m>n; \sup\limits_{x∈X} \left | \sum\limits_{i=n}^m f_i (x) \right | > ε] ⇔ [∃ε>0;∀N∈ℕ;∃n≥N;∃m>n;∃x∈X; \left | \sum\limits_{i=n}^m f_i (x) \right | ≥ ε]$ |
| $\left[ \lim\limits_{n⇝∞⁺} F_n (x) \mathop{\not↭}\limits_{x∈X} F (x) \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} | F_n (x) - F (x) | \mathop{\not↭}\limits_{x∈X} 0 \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} \sup\limits_{x∈X} | F_n (x) - F (x) | \not⇝ 0 \right]$ |
| $\left[ \sum\limits_{i=1}^{∞⁺} f_i (x) \mathop{\not↭}\limits_{x∈X} F (x) \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} \left | \sum\limits_{i=1}^n f_i (x) - F (x) \right | \mathop{\not↭}\limits_{x∈X} 0 \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} \sup\limits_{x∈X} \left | \sum\limits_{i=1}^n f_i (x) - F (x) \right | \not⇝ 0 \right]$ |
若通项不一致收敛于零，则函数项级数非一致收敛。
| $\lim\limits_{n⇝∞⁺} F_n (x) \mathop{\not↭}\limits_{x∈X} F (x) ⇐ \lim\limits_{i⇝∞⁺} f_i (x) \mathop{\not↭}\limits_{x∈X} 0 ⇔ \lim\limits_{i⇝∞⁺} \sup\limits_{x∈X} | f_i (x) | \not⇝ 0 $ |
典例：函数项级数$F_n (x) = x^0 + \sum\limits_{i=1}^n (x^i - x^{i-1}) = x^n$在区间$(0, 1)$上非一致收敛。
$F (x) = \lim\limits_{n⇝∞⁺} F_n (x) = \lim\limits_{n⇝∞⁺} x^n \mathop{⇝}\limits_{x∈(0, 1)} 0$
| $\lim\limits_{n⇝∞⁺} \sup\limits_{x∈(0, 1)} | F_n (x) - F(x) | = \lim\limits_{n⇝∞⁺} \sup\limits_{x∈(0, 1)} | x^n | ⇝ 1 ≠ 0$ |
若函数项级数有优正项级数，则必定绝对一致收敛。
若函数项级数绝对一致收敛，则必定条件一致收敛。
| $\left[ ∃N∈ℕ;∀i≥N; \sup\limits_{x∈X} | f_i (x) | ≤ | s_i | \right] ⇔ \left[ ∃N∈ℕ;∀i≥N;∀x∈X; | f_i (x) | ≤ | s_i | \right] ⇒ \left[ \sum\limits_{i=1}^{∞⁺} | s_i | ⇝ S ⇒ \sum\limits_{i=1}^{∞⁺} | f_i (x) | \mathop{↭}\limits_{x∈X} F_{ | } (x) ⇒ \sum\limits_{i=1}^{∞⁺} f_i (x) \mathop{↭}\limits_{x∈X} F (x) \right]$ |
| $\left[ ∃N∈ℕ;∀n≥N;∀m>n; \sup\limits_{x∈X} \left | \sum\limits_{i=n}^m f_i (x) \right | ≤ \sup\limits_{x∈X} \left | \sum\limits_{i=n}^m | f_i (x) | \right | ≤ \left | \sum\limits_{i=n}^m | s_i | \right | < ε \right] ⇔ \left[ ∃N∈ℕ;∀n≥N;∀m>n;∀x∈X; \left | \sum\limits_{i=n}^m f_i (x) \right | ≤ \left | \sum\limits_{i=n}^m | f_i (x) | \right | ≤ \left | \sum\limits_{i=n}^m | s_i | \right | < ε \right]$ |
函数项级数一致收敛的性质
若函数项级数收敛，且函数项为连续函数，则其和函数未必为连续函数。
典例：函数项级数$\sum\limits_{i=\rlap{≡}{0,}1}^{∞⁺} \dfrac{\sin i · x}{i} = \dfrac{\pi - x}{2}$，以$T = 2 · \pi$为周期，在点$x = 2 · i · \pi$处不连续。
若函数项级数一致收敛，且函数项为连续函数，则其和函数必定为连续函数。$P ∧ Q ⇒ R$
若函数项为连续函数，但和函数非连续函数，则函数项级数必定非一致收敛。$Q ∧ ¬R ⇒ ¬P$
| $\fbox{1}$ | $∀ε_1>0;∃δ>0;∃N∈ℕ;∀x∈X;∀n≥N; x_0∈X ∧ | x - x_0 | < δ ⇒ | F_n (x) - F_n (x_0) | < ε_1$ | ||||
|---|---|---|---|---|---|---|---|---|---|
| $\fbox{2}$ | $∀ε_3>0;∃N∈ℕ;∀n≥N;∀x∈X; | F_n (x) - F (x) | < ε_3$ | ||||||
| $⇓$ | $∀ε_2>0;∃N∈ℕ;∀n≥N; x_0∈X ⇒ | F_n (x_0) - F (x_0) | < ε_2$ | ||||||
| $⇓$ | $∀ε_1>0;∀ε_2>0;∀ε_4=ε_1+ε_2;∃δ>0;∃N∈ℕ;∀x∈X;∀n≥N; x_0∈X ∧ | x - x_0 | < δ ⇒ | F_n (x) - F (x_0) | < | F_n (x) - F_n (x_0) | + | F_n (x_0) - F (x_0) | < ε_4$ |
| $⇓$ | $∀ε_4>0;∃δ>0;∃N∈ℕ;∀x∈X;∀n≥N; x_0∈X ∧ | x - x_0 | < δ ⇒ | F_n (x) - F (x_0) | < ε_4$ | ||||
| $⇕$ | $∀ε_4>0;∃δ>0;∃N∈ℕ;∀n≥N;∀x∈X; x_0∈X ∧ | x - x_0 | < δ ⇒ | F_n (x) - F (x_0) | < ε_4$ | ||||
| $⇓$ | $\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} F_n (x) \mathop{≡}\limits_{\fbox{1} } \lim\limits_{n⇝∞⁺ \ x⇝x_0} F_n (x) \mathop{≡}\limits_{\fbox{2} } \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝x_0} F_n (x)$ | ||||||||
| $⇓$ | $∀ε_3>0;∀ε_4>0;∀ε_5=ε_3+ε_4;∃δ>0;∃N∈ℕ;∀n≥N;∀x∈X; x_0∈X ∧ | x - x_0 | < δ ⇒ | F (x) - F (x_0) | ≤ | F (x) - F_n (x) | + | F_n (x) - F (x_0) | < ε_5$ |
| $⇓$ | $∀ε_5>0;∃δ>0;∀x∈X; x_0∈X ∧ | x - x_0 | < δ ⇒ | F (x) - F (x_0) | < ε_5$ | ||||
| $⇓$ | $\lim\limits_{x⇝x_0} F (x) ⇝ F (x_0)$ |
若函数项级数$F_n (x)$在区间$X$上一致收敛，则多重极限可以随意交换极限运算的次序。
$\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} F_n (x) ≡ \lim\limits_{n⇝∞⁺ \ x⇝x_0} F_n (x) ≡ \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝x_0} F_n (x)$
若函数项级数$F_n (x)$在区间$X$上非一致收敛，则多重极限不可随意交换极限运算的次序。
$\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} F_n (x) \not≡ \lim\limits_{n⇝∞⁺ \ x⇝x_0} F_n (x) \not≡ \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝x_0} F_n (x)$
典例：分段函数项级数$F_n (x) = \mathop{2 · n · x}\limits_{0 ≤ x ≤ \frac{1}{2 · n} }; \mathop{2 · (1 - n · x)}\limits_{\frac{1}{2 · n} < x ≤ \frac{1}{n} }; \mathop{0}\limits_{\frac{1}{n} < x ≤ 1}$在区间$[0, 1]$内非一致收敛。
$\lim\limits_{x⇝0} \lim\limits_{n⇝∞⁺} F_n (x) = \lim\limits_{x⇝0} 0 ⇝ 0$
$\lim\limits_{n⇝∞⁺ \ x≡\frac{1}{2 · n}⇝0} F_n (x) = \lim\limits_{n⇝∞⁺} 2 · n · \dfrac{1}{2 · n} ⇝ 1 $
$\lim\limits_{n⇝∞⁺} \lim\limits_{x⇝0} F_n (x) = \lim\limits_{n⇝∞⁺} 0 ⇝ 0 $
函数项级数不可随意交换极限运算与求和运算的次序，除非函数项级数$F_n (x)$在区间$X$上一致收敛。
$\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} F_n (x) ≡ \lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^{n} f_i (x) ≡ \lim\limits_{x⇝x_0} \sum\limits_{i=0}^{∞⁺} f_i (x) \not≡ \sum\limits_{i=0}^{∞⁺} \lim\limits_{x⇝x_0} f_i (x) ≡ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n \lim\limits_{x⇝x_0} f_i (x) ≡ \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝x_0} \sum\limits_{i=0}^n f_i (x) ≡ \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝x_0} F_n (x)$
函数项级数不可随意交换极限运算与积分运算的次序，除非函数项级数$F_n (x)$在区间$[α, β]$上一致收敛。？
$\int\limits_α^β \left[ \lim\limits_{n⇝∞⁺} F_n (x) \right] \mathrm{d}x ≡ \int\limits_α^β \left[ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n f_i (x) \right] \mathrm{d}x ≡ \int\limits_α^β \left[ \sum\limits_{i=0}^{∞⁺} f_i (x) \right] \mathrm{d}x \not≡ \sum\limits_{i=0}^{∞⁺} \left[ \int\limits_α^β f_i (x) \mathrm{d}x \right] ≡ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n \left[ \int\limits_α^β f_i (x) \mathrm{d}x \right] ≡ \lim\limits_{n⇝∞⁺} \int\limits_α^β \left[ \sum\limits_{i=0}^n f_i (x) \right] \mathrm{d}x ≡ \lim\limits_{n⇝∞⁺} \int\limits_α^β F_n (x) \mathrm{d}x$
函数项级数不可随意交换极限运算与求导运算的次序，除非函数项级数$F_n (x)$在区间$X$上一致收敛。？
$\lim\limits_{x⇝x_0} \dfrac{\mathrm{d} }{\mathrm{d}x} \left[ \lim\limits_{n⇝∞⁺} F_n (x) \right] ≡ \lim\limits_{x⇝x_0} \dfrac{\mathrm{d} }{\mathrm{d}x} \left[ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n f_i (x) \right] ≡ \lim\limits_{x⇝x_0} \dfrac{\mathrm{d} }{\mathrm{d}x} \left[ \sum\limits_{i=0}^{∞⁺} f_i (x) \right] \not≡ \sum\limits_{i=0}^{∞⁺} \lim\limits_{x⇝x_0} \dfrac{\mathrm{d} f_i (x)}{\mathrm{d}x} ≡ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n \lim\limits_{x⇝x_0} \dfrac{\mathrm{d} f_i (x)}{\mathrm{d}x} ≡ \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝x_0} \dfrac{\mathrm{d} }{\mathrm{d}x} \left[ \sum\limits_{i=0}^n f_i (x) \right] ≡ \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝x_0} \dfrac{\mathrm{d} F_n (x)}{\mathrm{d}x}$
典例：函数项级数$F_n (x) = \sum\limits_{i=\rlap{≡}{0,}1}^n \left[ i · x · (1 - x^2)^i - (i - 1) · x · (1 - x^2)^{i-1} \right] $在区间$[0, 1]$上非一致收敛。
$\int\limits_0^1 \left[ \lim\limits_{n⇝∞⁺} F_n (x) \right] \mathrm{d}x = \int_0^1 \lim\limits_{n⇝∞⁺} \left[ n · x · (1 - x^2)^n \right] \mathrm{d}x = \int\limits_0^1 \lim\limits_{n⇝∞⁺} \left[ \mathop{0}\limits_{x=0,1}; \mathop{n · x · (1 - x^2)^n}\limits_{x∈(0, 1)} \right] \mathrm{d}x = \int\limits_0^1 0 \mathrm{d}x = 0 ≠ \dfrac{1}{2}$
| $\lim\limits_{n⇝∞⁺} \int\limits_0^1 F_n (x) \mathrm{d}x = \lim\limits_{n⇝∞⁺} \int\limits_0^1 [ n · x · (1 - x^2)^n ] \mathrm{d}x = \lim\limits_{n⇝∞⁺} \left. \dfrac{- n · (1 - x^2)^{n + 1} }{2 · (n + 1)} \right | 0^1 = \lim\limits{n⇝∞⁺} \dfrac{n}{2 · (n + 1)} ⇝ \dfrac{1}{2}$ |
典例：函数项级数$F_n (x) = \sum\limits_{i=\rlap{≡}{0,}1}^n f_i (x) = \sum\limits_{i=\rlap{≡}{0,}1}^n \left[ ә^{-i · x^2} - ә^{-(i - 1) · x^2} \right] = ә^{-n · x^2} - 1$
$\lim\limits_{x⇝0} \dfrac{\mathrm{d} }{\mathrm{d}x} \left[ \lim\limits_{n⇝∞⁺} F_n (x) \right] = \lim\limits_{x⇝0} \dfrac{\mathrm{d} }{\mathrm{d}x} \left[ \lim\limits_{n⇝∞⁺} \left( ә^{-n·x^2} - 1 \right) \right] = \lim\limits_{x⇝0} \dfrac{\mathrm{d} }{\mathrm{d}x} \left[ \mathop{0}\limits_{x=0}, \mathop{-1}\limits_{x≠0} \right] \not≡ 0$
$\lim\limits_{n⇝∞⁺} \lim\limits_{x⇝0} \dfrac{\mathrm{d} F_n (x)}{\mathrm{d}x} = \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝0} \dfrac{\mathrm{d} (ә^{-n · x^2} - 1)}{\mathrm{d}x} = \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝0} -2 · n · x · ә^{-n · x^2} ≡ 0$
幂级数
幂级数的部分和
$F_n (x) ≡ \sum\limits_{i=0}^n p_i · (x - x_0)^i$
幂级数的极限
$\lim\limits_{n⇝∞⁺} F_n (x) ≡ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n p_i · (x - x_0)^i ≡ \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0)^i$
幂级数收敛
| $\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} F_n (x) ⇝ F (x_0) ⇔ \lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} | F_n (x) - F (x_0) | ⇝ 0$ |
| $\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n p_i · (x - x_0)^i ⇝ F (x_0) ⇔ \lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} \left | \sum\limits_{i=0}^n p_i · (x - x_0)^i - F (x_0) \right | ⇝ 0$ |
幂级数发散
| $\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} F_n (x) \not⇝ F (x_0) ⇔ \lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} | F_n (x) - F (x_0) | \not⇝ 0$ |
| $\lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n p_i · (x - x_0)^i \not⇝ F (x_0) ⇔ \lim\limits_{x⇝x_0} \lim\limits_{n⇝∞⁺} \left | \sum\limits_{i=0}^n p_i · (x - x_0)^i - F (x_0) \right | \not⇝ 0$ |
幂级数收敛的性质
若幂级数在点$x_1 - x_0$处条件收敛，则在区间$[±0, x_1 - x_0]$上条件一致收敛。
若幂级数在点$x_1 - x_0$处绝对收敛，则在区间$[±0, x_1 - x_0]$上绝对一致收敛。
$\left[ \sum\limits_{i=0}^{∞⁺} p_i · (x_1 - x_0)^i ⇝ F (x_1) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0)^i \mathop{↭}\limits_{[±0, x - x_0]⊆[±0, x_1 - x_0]} F (x) \right]$
| $\left[ \sum\limits_{i=0}^{∞⁺} | p_i · (x_1 - x_0)^i | ⇝ F_{ | } (x_1) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} | p_i · (x - x_0)^i | \mathop{↭}\limits_{[±0, x - x_0]⊆[±0, x_1 - x_0]} F_{ | } (x) \right]$ |
若幂级数在点$x_1 - x_0$处条件收敛，则在区间$[±0, x_1 - x_0)$上绝对收敛，且在区间$[±0, x_1 - x_0]$上条件收敛。
若幂级数在点$x_1 - x_0$处绝对收敛，则在区间$[±0, x_1 - x_0]$上绝对收敛，且在区间$[±0, x_1 - x_0]$上条件收敛。
若幂级数在点$x_2 - x_0$处绝对发散，则在区间$(x_2 - x_0, ±∞)$上条件发散，且在区间$[x_2 - x_0, ±∞)$上绝对发散。
若幂级数在点$x_2 - x_0$处条件发散，则在区间$[x_2 - x_0, ±∞)$上条件发散，且在区间$[x_2 - x_0, ±∞)$上绝对发散。
| $\left[ \sum\limits_{i=0}^{∞⁺} | p_i · (x_1 - x_0)^i | ⇝ F_{ | } (x_1) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} p_i · (x_1 - x_0)^i ⇝ F (x_1) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} | p_i · (x_2 - x_0)^i | \mathop{⇝}\limits_{ | x_2 - x_0 | < | x_1 - x_0 | } F_{ | } (x_2) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0)^i \mathop{⇝}\limits_{ | x - x_0 | ≤ | x_2 - x_0 | < | x_1 - x_0 | } F (x) \right]$ |
| $\left[ \sum\limits_{i=0}^{∞⁺} p_i · (x_2 - x_0)^i \not⇝ F (x_2) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} | p_i · (x_2 - x_0)^i | \not⇝ F_{ | } (x_2) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} p_i · (x_1 - x_0)^i \mathop{\not⇝}\limits_{ | x_1 - x_0 | > | x_2 - x_0 | } F (x_1) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} | p_i · (x - x_0)^i | \mathop{\not⇝}\limits_{ | x - x_0 | ≥ | x_1 - x_0 | > | x_2 - x_0 | } F_{ | } (x) \right]$ |
| $⇓$ | $\sum\limits_{i=1}^{∞⁺} p_i · (x - x_0)^i = \sum\limits_{i=1}^{∞⁺} p_i · (x_1 - x_0)^i · \dfrac{(x - x_0 )^i}{(x_1 - x_0)^i}$ | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $⇓$ | $\left[ \sum\limits_{i=1}^{∞⁺} p_i · (x_1 - x_0)^i ⇝ F (x_1) \right] ⇒ \left[ \sum\limits_{i=1}^{∞⁺} p_i · (x - x_0)^i \mathop{↭}\limits_{[±0, x - x_0] ⊆ [±0, x_1 - x_0]} F (x) \right]$ | $⇐$ | $\left[ \sum\limits_{i=1}^{∞⁺} p_i · (x_1 - x_0)^i ⇝ F (x_1) \right] ∧ \left[ \left | \dfrac{(x - x_0)^i}{(x_1 - x_0)^i} \right | ≤ 1 \right]$ | ||||||||||||
| $⇓$ | $\left[ \sum\limits_{i=1}^{∞⁺} | p_i · (x_1 - x_0)^i | ⇝ F_{ | } (x_1) \right] ⇒ \left[ \sum\limits_{i=1}^{∞⁺} | p_i · (x - x_0)^i | \mathop{↭}\limits_{[±0, x - x_0] ⊆ [±0, x_1 - x_0]} F_{ | } (x) \right]$ | $⇐$ | $\left[ \sum\limits_{i=1}^{∞⁺} | p_i · (x_1 - x_0)^i | ⇝ F_{ | } (x_1) \right] ∧ \left[ \left | \dfrac{(x - x_0)^i}{(x_1 - x_0)^i} \right | ≤ 1 \right]$ | |||
| $⇓$ | $\left[ \sum\limits_{i=1}^{∞⁺} p_i · (x_1 - x_0)^i ⇝ F (x_1) \right] ⇒ \left[ \sum\limits_{i=1}^{∞⁺} | p_i · (x - x_0)^i | \mathop{⇝}\limits_{ | x - x_0 | < | x_1 - x_0 | } F_{ | } (x) \right]$ | $⇐$ | $[ | p_i · (x_1 - x_0)^i | ≤ \mathrm{Sup.} ] ∧ \left[ \sum\limits_{i=1}^{∞⁺} \dfrac{(x - x_0)^i}{(x_1 - x_0)^i} ⇝ \mathrm{Con.} \right]$ | |||||
| $⇓$ | $\left[ \sum\limits_{i=1}^{∞⁺} | p_i · (x - x_0)^i | \not⇝ F_{ | } (x) \right] ⇒ \left[ \sum\limits_{i=1}^{∞⁺} p_i · (x_1 - x_0)^i \mathop{\not⇝}\limits_{ | x - x_0 | < | x_1 - x_0 | } F (x_1) \right]$ |
幂级数的绝对收敛半径$R$与绝对发散半径$\overline{R}$。
| $R ≡ \dfrac{1}{\varlimsup\limits_{i⇝∞⁺} \sqrt[i]{ | p_i | } } ≡ \varliminf\limits_{i⇝∞⁺} \dfrac{1}{\sqrt[i]{ | p_i | } } < \varlimsup\limits_{i⇝∞⁺} \dfrac{1}{\sqrt[i]{ | p_i | } } ≡ \dfrac{1}{\varliminf\limits_{i⇝∞⁺} \sqrt[i]{ | p_i | } } ≡ \overline{R}$ |
| 幂级数在绝对收敛半径内区间$ | x - x_0 | ≤ R_γ < R$上绝对一致收敛。 |
幂级数在绝对收敛半径内绝对收敛且条件收敛，幂级数在绝对发散半径外条件发散且绝对发散。
| $\left[ \sum\limits_{i=0}^{∞⁺} | p_i · (x - x_0 )^i | \mathop{↭}\limits_{ | x - x_0 | ≤ R_γ < R} F_{ | } (x) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0 )^i \mathop{↭}\limits_{ | x - x_0 | ≤ R_γ < R} F (x) \right]$ |
| $\left[ \sum\limits_{i=0}^{∞⁺} | p_i · (x - x_0)^i | \mathop{⇝}\limits_{ | x - x_0 | < R} F_{ | } (x) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0)^i \mathop{⇝}\limits_{ | x - x_0 | < R} F (x) \right]$ |
| $\left[ \sum\limits_{i=0}^{∞⁺} | p_i · (x - x_0)^i | \mathop{\not⇝}\limits_{ | x - x_0 | > \overline{R} } F_{ | } (x) \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0)^i \mathop{\not⇝}\limits_{ | x - x_0 | > \overline{R} } F (x) \right]$ |
| $⇑$ | $\varlimsup\limits_{i⇝∞⁺} | p_i · (x - x_0)^i | ⇝ \dfrac{1}{γ_+^i}$ | $\mathop{⇒}\limits^{γ_+ > 1}$ | $\sum\limits_{i=1}^{∞⁺} | p_i · (x - x_0 )^i | ↭ F_{ | } (x)$ | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $⇑$ | $\varliminf\limits_{i⇝∞⁺} \dfrac{1}{\sqrt[i]{ | p_i · (x - x_0)^i | } } = \varliminf\limits_{i⇝∞⁺} \dfrac{1}{\sqrt[i]{ | p_i | } · | x - x_0 | } ⇝ γ_+ > 1$ | $\mathop{⇒}\limits^{γ_+ > 1}$ | $\sum\limits_{i=1}^{∞⁺} | p_i · (x - x_0)^i | ⇝ F_{ | } (x)$ | |||||||
| $⇑$ | $\varlimsup\limits_{i⇝∞⁺} \dfrac{1}{\sqrt[i]{ | p_i · (x - x_0)^i | } } = \varlimsup\limits_{i⇝∞⁺} \dfrac{1}{\sqrt[i]{ | p_i | } · | x - x_0 | } ⇝ γ_- < 1$ | $\mathop{⇒}\limits^{γ_- < 1}$ | $\sum\limits_{i=1}^{∞⁺} | p_i · (x - x_0)^i | \not⇝ F_{ | } (x)$ | |||||||
| $⇑$ | $R ≡ \dfrac{1}{\varlimsup\limits_{i⇝∞⁺} \sqrt[i]{ | p_i | } } ≡ \varliminf\limits_{i⇝∞⁺} \dfrac{1}{\sqrt[i]{ | p_i | } } ⇝ γ_+ · | x - x_0 | > | x - x_0 | $ | $\overline{R} ≡ \dfrac{1}{\varliminf\limits_{i⇝∞⁺} \sqrt[i]{ | p_i | } } ≡ \varlimsup\limits_{i⇝∞⁺} \dfrac{1}{\sqrt[i]{ | p_i | } } ⇝ γ_- · | x - x_0 | < | x - x_0 | $ | |
| $⇑$ | $R_γ ≡ \dfrac{R}{γ_+} ≥ | x - x_0 | $ | $\mathop{⇒}\limits^{γ_+ > 1}$ | $\sum\limits_{i=1}^{∞⁺} | p_i · (x - x_0 )^i | \mathop{↭}\limits_{ | x - x_0 | ≤ R_{γ} < R} F_{ | } (x)$ | |||||||||
| $⇑$ | $R > | x - x_0 | $ | $⇒$ | $\sum\limits_{i=1}^{∞⁺} | p_i · (x - x_0)^i | \mathop{⇝}\limits_{ | x - x_0 | < R} F_{ | } (x)$ | |||||||||
| $⇑$ | $\overline{R} < | x - x_0 | $ | $\mathop{⇒}\limits^{\overline{R} ≥ R}$ | $\sum\limits_{i=1}^{∞⁺} | p_i · (x - x_0)^i | \mathop{\not⇝}\limits_{ | x - x_0 | > \overline{R} } F_{ | } (x)$ |
比值审敛法的绝对收敛半径小于等于根值审敛法的绝对收敛半径，比值审敛法的绝对发散半径大于等于根值审敛法的绝对发散半径。
| $\left[ \varliminf\limits_{i ⇝ ∞⁺} \left | \dfrac{p_i}{p_{i+1} } \right | ≤ \varliminf\limits_{i ⇝ ∞⁺} {\dfrac{1}{\sqrt[i]{ | p_i | } } } ≡ R \right] ≤ \left[ \overline{R} ≡ \varlimsup\limits_{i ⇝ ∞⁺} \dfrac{1}{\sqrt[i]{ | p_i | } } ≤ \varlimsup\limits_{i ⇝ ∞⁺} \left | \dfrac{p_i}{p_{i+1} } \right | \right]$ |
特例：幂级数$\sum\limits_{i=0 \ j=2·i,2·i+1}^{∞⁺} \left[ \dfrac{1}{2^{2 · i} } · (x - x_0)^{2·i} + \dfrac{1}{3^{2 · i + 1} } · (x - x_0)^{2·i+1} \right]$
| $\varliminf\limits_{j⇝∞⁺} \dfrac{ | p_j | }{ | p_{j+1} | } = \min\left\lbrace \lim\limits_{i⇝∞⁺} \dfrac{1/2^{2·i} }{1/3^{2·i+1} }, \lim\limits_{i⇝∞⁺} \dfrac{1/3^{2·i+1} }{1/2^{2·i+2} } \right\rbrace = \lim\limits_{i⇝∞⁺} \dfrac{2^{2·i+2} } {3^{2·i+1} } ⇝ 0$ |
| $\varlimsup\limits_{j⇝∞⁺} \dfrac{ | p_j | }{ | p_{j+1} | } = \max\left\lbrace \lim\limits_{i⇝∞⁺} \dfrac{1/2^{2·i} }{1/3^{2·i+1} }, \lim\limits_{i⇝∞⁺} \dfrac{1/3^{2·i+1} }{1/2^{2·i+2} } \right\rbrace = \lim\limits_{i⇝∞⁺} \dfrac{3^{2·i+1} }{2^{2·i} } ⇝ ∞⁺$ |
| $\varliminf\limits_{j⇝∞⁺} \dfrac{1}{\sqrt[j]{ | p_j | } } = \min\left\lbrace \lim\limits_{i⇝∞⁺} \sqrt[2·i]{2^{2·i} }, \lim\limits_{i⇝∞⁺} \sqrt[2·i+1]{3^{2·i+1} } \right\rbrace = \lim\limits_{i⇝∞⁺} \sqrt[2·i]{2^{2·i} } = 2 > 0$ |
| $\varliminf\limits_{j⇝∞⁺} \dfrac{1}{\sqrt[j]{ | p_j | } } = \max\left\lbrace \lim\limits_{i⇝∞⁺} \sqrt[2·i]{2^{2·i} }, \lim\limits_{i⇝∞⁺} \sqrt[2·i+1]{3^{2·i+1} } \right\rbrace = \lim\limits_{i⇝∞⁺} \sqrt[2·i+1]{3^{2·i+1} } = 3 < ∞⁺$ |
| 幂级数在绝对收敛半径内区间$ | x - x_0 | < R$上，连续且有任意阶导数。 |
| 幂级数在绝对收敛半径内区间$ | x - x_0 | ≤ R_γ < R$上，可逐项积分且积分后绝对收敛半径不变，绝对收敛半径处的敛散性可能由发散变成收敛。 |
| 幂级数在绝对收敛半径内区间$ | x - x_0 | ≤ R_γ < R$上，可逐项求导且求导后绝对收敛半径不变，绝对收敛半径处的敛散性可能由收敛变成发散。 |
$\int\limits_{x_0}^{x} \mathrm{d} x · \lim\limits_{n⇝∞⁺} F_n (x) ≡ \int\limits_{x_0}^{x} \mathrm{d} x · \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0)^i ≡ \sum\limits_{i=0}^{∞⁺} \int\limits_{x_0}^{x}\mathrm{d} x · p_i · (x - x_0)^i ≡ \lim\limits_{n⇝∞⁺} \int\limits_{x_0}^{x} \mathrm{d} x · F_n (x) ≡ \sum\limits_{i=0}^{∞⁺} \dfrac{p_i}{i + 1} · (x - x_0)^{i + 1}$
$\dfrac{\mathrm{d} }{\mathrm{d} x} \lim\limits_{n⇝∞⁺} F_n (x) ≡ \dfrac{\mathrm{d} }{\mathrm{d} x} \sum\limits_{i=0}^{∞⁺} [ p_i · (x - x_0)^i ] ≡ \sum\limits_{i=0}^{∞⁺} \dfrac{\mathrm{d} }{\mathrm{d} x} [ p_i · (x - x_0)^i] ≡ \lim\limits_{n⇝∞⁺} \dfrac{\mathrm{d} }{\mathrm{d} x} F_n (x) ≡ \sum\limits_{i=0}^{∞⁺} [ p_i · i · (x - x_0)^{i - 1} ]$
| $\varliminf\limits_{n⇝∞⁺} \sqrt[i]{\left | \dfrac{p_i}{i + 1} \right | } = \dfrac{\varliminf\limits_{n⇝∞⁺} \sqrt[i]{ | p_i | } }{\varlimsup\limits_{n⇝∞⁺} \sqrt[i]{i + 1} } = R = \varliminf\limits_{n⇝∞⁺} \sqrt[i]{ | p_i · i | } = \varliminf\limits_{n⇝∞⁺} \sqrt[i]{ | p_i | } · \varliminf\limits_{n⇝∞⁺} \sqrt[i]{i} $ |
幂级数在绝对收敛半径$±R$处的敛散性不恒定。
典例：幂级数$\sum\limits_{i=0}^{∞⁺} (-x)^i = \dfrac{1}{1 + x}$的绝对收敛半径为$R = 1$。
$\lim\limits_{x⇝1^-} \sum\limits_{i=0}^{∞⁺} (-x)^i = \lim\limits_{x⇝1^-} \dfrac{1}{1 + x} ⇝ \dfrac{1}{2} \not⇜ \sum\limits_{i=0}^{∞⁺} (-1)^i = \sum\limits_{i=0}^{∞⁺} \lim\limits_{x⇝1^-} (-x)^i$
若幂级数在点$x_1 - x_0$处的绝对极限将收敛，则在点$x_1 - x_0$处绝对收敛为该极限。
| $\left[ \lim\limits_{x⇝x_1} \sum\limits_{i=0}^{∞⁺} | p_i · (x - x_0)^i | ⇝ F_{ | x_1 | } \right] \mathop{⇒}\limits^{ | x - x_0 | ≤ | x_1 - x_0 | } \left[ \sum\limits_{i=0}^{∞⁺} | p_i · (x_1 - x_0)^i | ⇝ F_{ | x_1 | } \right]$ |
| $F_{ | x_1 | } ⇜ \lim\limits_{x⇝x_1} \sum\limits_{i=0}^{∞⁺} | p_i | · | x - x_0 | ^i \mathop{≤}\limits^{ | x - x_0 | <≤ | x_1 - x_0 | } \sum\limits_{i=0}^{∞⁺} | p_i | · | x_1 - x_0 | ^i = \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n \lim\limits_{x⇝x_1} | p_i | · | x - x_0 | ^i = \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝x_1} \sum\limits_{i=0}^n | p_i | · | x - x_0 | ^i ≤ \rlap{≡≡≡} \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝x_1} \sum\limits_{i=0}^{∞⁺} | p_i | · | x - x_0 | ^i ⇝ F_{ | x_1 | }$ |
幂级数的运算性质
同起点幂级数可做加运算，其绝对收敛半径至少为共同的绝对收敛半径。
同起点幂级数可做乘运算，其绝对收敛半径至少为各绝对收敛半径之积。
| $\left[ \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0)^i \right] ± \left[ \sum\limits_{j=0}^{∞⁺} q_j · (x - x_0)^j \right] \mathop{=======}\limits_{ | x - x_0 | <R_{pq} }^{R_{pq}≡\min\lbrace R_p, R_q \rbrace} \sum\limits_{l=0}^{∞⁺} (p_l ± q_l) · (x - x_0)^l$ |
| $\left[ \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0)^i \right] · \left[ \sum\limits_{j=0}^{∞⁺} q_j · (x - x_0)^j \right] \mathop{=====}\limits_{ | x - x_0 | < R_{pq} }^{R_{pq} ≡ R_p · R_q} \sum\limits_{l=0}^{∞⁺} \left[ \sum\limits_{k=0}^l p_k · q_{l-k} \right] · (x - x_0)^l$ |
| $⇓$ | $\dfrac{1}{R_{pq} } ≡ \dfrac{1}{\min\lbrace R_p, R_q \rbrace} = \max\left\lbrace \dfrac{1}{R_p}, \dfrac{1}{R_q} \right\rbrace = \max\left\lbrace \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | p_l | }, \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | q_l | } \right\rbrace = \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{\max\lbrace | p_l | , | q_l | \rbrace}$ | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $⇓$ | $\dfrac{1}{R} ≡ \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | p_l ± q_l | } ≤ \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | p_l | + | q_l | } ≤ \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{2 · \max\lbrace | p_l | , | q_l | \rbrace} = \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{\max\lbrace | p_l | , | q_l | \rbrace} = \dfrac{1}{R_{pq} }$ | $⇒$ | $R_{pq} ≤ \dfrac{1}{\varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | p_l ± q_l | } } ≡ R$ | ||||||||
| $⇓$ | $\dfrac{1}{R_{pq} } = \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{\max\lbrace | p_l | , | q_l | \rbrace} ≤ \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | p_l | + | q_l | } \mathop{=====}\limits^{0 ≤ ± · p_l · q_l} \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | p_l ± q_l | } ≡ \dfrac{1}{R}$ | $⇒$ | $R_{pq} \mathop{=====}\limits^{0 ≤ ± · p_l · q_l} R$ | ||||||||||||||
| $⇓$ | $\left | \dfrac{1}{R_p} - \dfrac{1}{R_q} \right | = \left | \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | p_l | } - \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | q_l | } \right | ≤ \varlimsup\limits_{l⇝∞⁺} \left | \sqrt[l]{ | p_l | } - \sqrt[l]{ | q_l | } \right | ≤ \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | p_l | - | q_l | } ≤ \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{ | p_l ± q_l | } ≡ \dfrac{1}{R}$ | $⇒$ | $R_{pq} ≤ R ≤ \dfrac{R_p · R_q}{ | R_p - R_q | } \mathop{========}\limits^{\max\lbrace R_p, R_q \rbrace = ∞⁺} R_{pq}$ |
| $⇓$ | $\left[ \lim\limits_{n⇝∞^{+} } s_n · t_n \mathop{⇝}\limits_{\lim\limits_{n⇝∞^{+} } t_n ⇝ t,∞^{±} }^{\lim\limits_{n⇝∞^{+} } s_n ⇝ s,∞^{±} } r,∞^{±} \right] ⇒ \left[ \lim\limits_{n⇝∞^{+} } \dfrac{1}{n + 1} · \sum\limits_{i=0}^{n} s_i · t_{n-i} ⇝ r,∞^{±} \right]$ | $\lim\limits_{n⇜∞^{+} } \sqrt[l+1]{l + 1} ⇝ 1$ | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $⇓$ | $\dfrac{1}{R} ≡ \varlimsup\limits_{l⇝∞⁺} \sqrt[l+1]{\left | \sum\limits_{k=0}^l p_k · q_{l-k} \right | } ≤ \varlimsup\limits_{l⇝∞⁺} \sqrt[l+1]{\sum\limits_{k=0}^l | p_k | · | q_{l-k} | } = \varlimsup\limits_{l⇝∞⁺} \sqrt[l+1]{l + 1} · \varlimsup\limits_{l⇝∞⁺} \sqrt[l+1]{\dfrac{1}{l + 1} · \sum\limits_{k=0}^l | p_k | · | q_{l-k} | } = \varlimsup\limits_{l⇝∞⁺} \sqrt[l+1]{ | p_l | · | q_l | } ⇝ \dfrac{1}{R_p · R_q}$ | $⇒$ | $R_p · R_q ≤ R$ |
| $⇓$ | $\dfrac{1}{R_p · R_q} = \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{\sum\limits_{k=0}^l | p_k | · | q_{l-k} | } \mathop{===}\limits_{q_j⪌0}^{p_i⪌0} \varlimsup\limits_{l⇝∞⁺} \sqrt[l]{\left | \sum\limits_{k=0}^l p_k · q_{l-k} \right | } ≡ \dfrac{1}{R}$ | $⇒$ | $R_p · R_q \mathop{===}\limits_{q_j⪌0}^{p_i⪌0} R$ |
| 附加证明：$ | p_l | ^{\frac{1}{l} } - | q_l | ^{\frac{1}{l} } | ≤ | p_l | - | q_l | ^\frac{1}{l}$ |
| $⇓$ | $ | x | + | y | ≤ ( | x | ^\frac{1}{l} + | y | ^{\frac{1}{l} })^l$ | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $⇓$ | $( | x | + | y | )^\frac{1}{l} ≤ | x | ^\frac{1}{l} + | y | ^{\frac{1}{l} }$ | ||||
| $⇓$ | $ | x | ^{\frac{1}{l} } ≤ | x | - | y | ^{\frac{1}{l} } + | y | ^{\frac{1}{l} }$ | ||||
| $⇓$ | $ | x | ^{\frac{1}{l} } - | y | ^{\frac{1}{l} } | ≤ | x | - | y | ^\frac{1}{l}$ |
同起点幂级数与等比级数之积，其绝对收敛半径至少为原幂级数的绝对收敛半径。
$\dfrac{1}{1 - (x - x_0)} · \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0)^i = \sum\limits_{j=0}^{∞⁺} (x - x_0)^j · \sum\limits_{i=0}^{∞⁺} p_i · (x - x_0)^i = \sum\limits_{l=0}^{∞⁺} \left[ \sum\limits_{k=0}^l p_k \right] · (x - x_0)^l = \sum\limits_{l=0}^{∞⁺} P_l · (x - x_0)^l$
幂级数的无穷阶展开
若函数$f (x)$在点$x = x_0$处连续且有无穷阶导数，则在点$x = x_0$处可唯一展开成无穷阶幂级数。
若函数$f (x)$在点$x = x_0$处展开成无穷阶幂级数，则在点$x = x_0$处收敛于自身，但在点$x ≠ x_0$处未必收敛或未必收敛于自身。
$f (x) = \lim\limits_{n⇝∞⁺} \left[ \sum\limits_{i=0}^n \dfrac{ {^i}f (x_0)}{i!} · (x - x_0)^i = f (x_0) + \dfrac{ {^1}f (x_0)}{1!} · (x - x_0)^1 + \dfrac{ {^2}f (x_0)}{2!} · (x - x_0)^2 + ··· + \dfrac{ {^n}f (x_0)}{n!} · (x - x_0)^n + R_n (x) \right]$
| $f (x) = \lim\limits_{n⇝∞⁺} \left[ \sum\limits_{i=0}^n \left.\dfrac{\mathrm{d}^i f (x)}{i! · \mathrm{d}^i x}\right | _{x_0} · (x - x_0)^i = f (x_0) + \left.\dfrac{\mathrm{d}^1 f (x)}{1! · \mathrm{d}^1 x}\right | _{x_0} · (x - x_0)^1 + \left.\dfrac{\mathrm{d}^2 f (x)}{2! · \mathrm{d}^2 x}\right | _{x_0} · (x - x_0)^2 + ··· + \left.\dfrac{\mathrm{d}^n f (x)}{n! · \mathrm{d}^n x}\right | _{x_0} · (x - x_0)^n + R_n (x) \right]$ |
| $f (x) = \lim\limits_{n⇝∞⁺} \left[ \sum\limits_{i=0}^n \left.\dfrac{\mathrm{d}^i f (x)}{\mathrm{d} x^i}\right | _{x_0} · (x - x_0)^i = f (x_0) + \left.\dfrac{\mathrm{d}^1 f (x)}{\mathrm{d} x^1}\right | _{x_0} · (x - x_0)^1 + \left.\dfrac{\mathrm{d}^2 f (x)}{\mathrm{d} x^2}\right | _{x_0} · (x - x_0)^2 + ··· + \left.\dfrac{\mathrm{d}^n f (x)}{\mathrm{d} x^n}\right | _{x_0} · (x - x_0)^n + R_n (x) \right]$ |
$R_n (x) = o (x - x_0)^n = f (x) - f (x_0) - \sum\limits_{i=1}^n \dfrac{ {^i}f (x_0)}{i!} · (x - x_0)^i \mathop{====}\limits_{∃θ∈[x_0, x]} \dfrac{ {^{n+1} }f (θ)}{(n + 1)!} · (x - x_0)^{n + 1} \mathop{====}\limits_{∃θ∈[x_0, x]} \dfrac{ {^{n+1}f (θ)} }{n!} · (x - θ)^n · (x - x_0)^1 \mathop{====}\limits_{∃θ∈[x_0, x]} \int\limits_{x_0}^x \dfrac{ {^{n+1} }f (t)}{n!} · (x - t)^n \mathrm{d} t$
| $[ R_n (x_0) = 0 ] ⇒ [ f (x) | _{x_0} = f (x_0) ]$ |
有函数$f (x)$在点$x = x_0$处展开成无穷阶幂级数，若在点$x ≠ x_0$处收敛于自身，则在点$x ≠ x_0$处余项$R_n (x)$的极限为零，反之亦然。
| $\left[ f (x) = \sum\limits_{i=0}^{∞⁺} \dfrac{ {^i}f (x_0)}{i!} · (x - x_0)^i \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} | R_n (x) | ⇝ 0 \right]$ |
有函数$f (x)$在点$x = x_0$处展开成无穷阶幂级数，若在区间$[±x_0, x_1]$内各阶导数有确界，则在区间$[+x_0, x_1]$内收敛于自身，反之不对。
| $\left[ | {^i}f (θ) | ≤ \mathrm{Sup.} \right] ⇒ \left[ \lim\limits_{n⇝∞⁺} | R_n (x) | \mathop{====}\limits_{∃θ∈[x_0, x]} \lim\limits_{n⇝∞⁺} \left | \dfrac{ {^{n+1} }f (θ)}{(n + 1)!} · (x - x_0)^{n + 1} \right | ≤ \lim\limits_{n⇝∞⁺} \mathrm{Sup.} · \dfrac{ | (x - x_0)^{n + 1} | }{(n + 1)!} ⇝ 0 \right]$ |
有函数$f (x)$在点$x = x_0$处展开成无穷阶幂级数，若在区间$[+x_0, x_1]$内各阶导数非负数，则在区间$[+x_0, x_1)$内收敛于自身，反之不对。
| $⇓$ | $f (x) = \lim\limits_{n⇝∞⁺} \left[ \sum\limits_{i=0}^n \dfrac{ {^i}f (x_0)}{i!} · (x - x_0)^i + R_n (x) \right]$ | $⇒$ | $\left[ 0 ≤ {^i} f (x) \right] \mathop{⇒}\limits_{x∈[+x_0, x_1]} [ R_n (x) ≤ f (x)]$ |
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| $⇓$ | $\dfrac{R_n (x)}{(x - x_0)^{n + 1} } = \dfrac{ {^{n+1} }f (x_0)}{(n + 1)!} + \dfrac{ {^{n+2} }f (x_0)}{(n + 2)!} · (x - x_0)^{1} + \dfrac{ {^{n+3} }f (x_0)}{(n + 3)!} · (x - x_0)^2 + T_{n+3} (x)$ | $⇒$ | $g (x) ≡ \dfrac{\mathrm{d} }{\mathrm{d} x} \left[ \dfrac{R_n (x)}{(x - x_0)^{n + 1} } \right] = g (x_0) + \dfrac{ {^1}g (θ)}{1！} · (x - x_0)^1$ |
| $⇓$ | $g (x) = \dfrac{\mathrm{d} }{\mathrm{d} x} \left[ \dfrac{R_n (x)}{(x - x_0)^{n + 1} } \right] \mathop{=====}\limits_{x∈[+x_0, x_1]}^{θ∈[+x_0, x]} \dfrac{ {^{n+2} }f (x_0)}{(n + 2)!} + \left[ \dfrac{ {^{n+3} }f (x_0)}{(n + 3)!} · 2 + {^2}T_{n+3} (θ) \right] · (x - x_0)^1 ≥ 0$ | $⇐$ | ${^1}g (x) = \dfrac{ {^{n+3} }f (x_0)}{(n + 3)!} · 2 + {^2}T_{n+3} (x) ≥ 0$ |
| $⇓$ | $\dfrac{R_n (x)}{(x - x_0)^{n + 1} } ≤ \dfrac{R_n (x_1)}{(x_1 - x_0)^{n + 1} } ≤ \dfrac{f (x_1)}{(x_1 - x_0)^{n + 1} }$ | ||
| $⇓$ | $\lim\limits_{n⇝∞⁺} R_n (x) ≤ \lim\limits_{n⇝∞⁺} \left[ \dfrac{x - x_0}{x_1 - x_0} \right]^{n + 1} · f (x_1) \mathop{⇝}\limits_{x∈[+x_0, x_1)} 0$ | $⇐$ | $\dfrac{x - x_0}{x_1 - x_0} \mathop{<}\limits_{x∈[+x_0, x_1)} 1$ |
| 无穷阶幂级数$f (x) = \sum\limits_{i=0}^{∞⁺} \left. \dfrac{\mathrm{d}^i f (x)}{i! · \mathrm{d}^i x} \right | {x_0} · (x - x_0)^i$，在绝对收敛半径$R{x_0}$内点$x = x_1$处可唯一展开成无穷阶幂级数，其绝对收敛半径至少为$R_{x_1} = R_{x_0} - | x_1 - x_0 | $。 |
| $⇓$ | $ | x - x_0 | = | x - x_1 + x_1 - x_0 | ≤ | x - x_1 | + | x_1 - x_0 | ≤ R_{x_0} $ | $⇔$ | $ | x - x_1 | ≤ R_{x_0} - | x_1 - x_0 | = R_{x_1}$ |
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| $⇓$ | $f_{x_0} (x) \mathop{=====}\limits_{ | x-x_1 | ≤R_{x_1} } \sum\limits_{i=0}^{∞⁺} \left. \dfrac{\mathrm{d}^i f (x)}{i! · \mathrm{d}^i x} \right | {x_0} · (x - x_0)^i = \sum\limits{i=0}^{∞⁺} \left. \dfrac{\mathrm{d}^i f (x)}{i! · \mathrm{d}^i x} \right | _{x_0} · [(x - x_1) + (x_1 - x_0)]^i$ | ||||||||||
| $⇓$ | $f_{x_0} (x) \mathop{=====}\limits_{ | x-x_1 | ≤R_{x_1} } \sum\limits_{i=0}^{∞⁺} \left[ \left. \dfrac{\mathrm{d}^{i-j} }{i! · \mathrm{d}^{i-j} x} \dfrac{\mathrm{d}^j f (x)}{\mathrm{d}^j x} \right | {x_0} · \sum\limits{j=0}^{i} \dfrac{i!}{j! · (i - j)!} · (x - x_1)^j · (x_1 - x_0)^{i - j} \right]$ | |||||||||||
| $⇓$ | $f_{x_1} (x) \mathop{=====}\limits_{ | x-x_1 | ≤R_{x_1} } \sum\limits_{j=0}^{∞⁺} \dfrac{1}{j!} · \left[ \sum\limits_{i=j}^{∞⁺} \dfrac{\mathrm{d}^{i-j} }{(i - j)! · \mathrm{d}^{i-j} x} \left. \dfrac{\mathrm{d}^j f (x)}{\mathrm{d}^j x} \right | _{x_0} · (x_1 - x_0)^{i-j} \right] · (x - x_1)^j$ | |||||||||||
| $⇓$ | $f_{x_1} (x) \mathop{=====}\limits_{ | x-x_1 | ≤R_{x_1} } \sum\limits_{j=0}^{∞⁺} \dfrac{1}{j!} · \left[ \sum\limits_{i=0}^{∞⁺} \dfrac{\mathrm{d}^{i} }{i! · \mathrm{d}^i x} \left. \dfrac{\mathrm{d}^j f (x)}{\mathrm{d}^j x} \right | _{x_0} · (x_1 - x_0)^{i} \right] · (x - x_1)^j$ | |||||||||||
| $⇓$ | $f_{x_1} (x) \mathop{=====}\limits_{ | x-x_1 | ≤R_{x_1} } \sum\limits_{j=0}^{∞⁺} \left. \dfrac{\mathrm{d}^j f (x)}{j! · \mathrm{d}^j x} \right | _{x_1} · (x - x_1)^j$ |
典例：函数$F (x) = \dfrac{1}{1 - x} = \sum\limits_{i=0}^{∞⁺} x^i$在点$x_0=0$处连续且有无穷阶导数。其在点$x_0=0$处展开的无穷阶幂级数在绝对发散半径$\overline{R} = 1$外条件发散。
典例：函数$F (x) = \sum\limits_{i=0}^{∞⁺} \dfrac{\sin 2^i · x}{i!}$在点$x_0 = 0$处连续且有无穷阶导数。其在点$x_0 = 0$处展开的无穷阶幂级数收敛于自身，但在点$x ≠ 0$处条件发散。
| $⇓$ | $\left[ \left | \dfrac{\sin 2^i · x}{i!} \right | \mathop{≤}\limits_{x∈(∞⁻, ∞⁺)} \dfrac{1}{i!} \mathop{<}\limits_{i⇝∞⁺} \dfrac{1}{2^i} \right] ⇒ \left[ \sum\limits_{i=0}^{∞⁺} \left | \dfrac{\sin 2^i · x}{i!} \right | \mathop{↭}\limits_{x∈(∞⁻, ∞⁺)} F_{ | } (x) \right]$ | ||
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| $⇓$ | $\dfrac{\mathrm{d}^j F (x)}{\mathrm{d}^j x} = \sum\limits_{i=0}^{∞⁺} \dfrac{\mathrm{d}^j}{\mathrm{d}^j x} \left[ \dfrac{\sin 2^i · x}{i!} \right] = \sum\limits_{i=0}^{∞⁺} \dfrac{(2^i)^j · \sin \left( 2^i · x + j · \dfrac{π}{2} \right)}{i!}$ | |||||||
| $⇓$ | $\left. \dfrac{\mathrm{d}^j F (x)}{j! · \mathrm{d}^j x} \right | {x_0=0} \mathop{====}\limits{j = 2 · k + 1}^{\rlap{≡≡≡}{j = 2 · k} } \sum\limits_{i=0}^{∞⁺} \dfrac{(2^i)^{2 · k +1} · \sin \left( k · π + \dfrac{π}{2} \right)}{(2 · k + 1)! · i!} = \sum\limits_{i=0}^{∞⁺} \dfrac{(-1)^k}{(2 · k + 1)!} · \dfrac{(2^{2 · k + 1})^i}{i!} = \dfrac{(-1)^k}{(2 · k + 1)!} · ә^{2^{2 · k + 1} }$ | $\left. \dfrac{\mathrm{d}^j F (x)}{j! · \mathrm{d}^j x} \right | {x_0} \mathop{===}\limits^{j=2·k} \sum\limits{i=0}^{∞⁺} \dfrac{(2^i)^{2 · k} · \sin\left( k · π \right)}{(2 · k)! · i!} = 0$ | ||||
| $⇓$ | $F (x) = \sum\limits_{j=2·k+1 \ k=0}^{∞⁺} \dfrac{(-1)^k · ә^{2^{2 · k + 1} } }{(2 · k + 1)!} · x^{2 · k + 1}$ | |||||||
| $⇓$ | $\overline{R}= \varlimsup\limits_{j=2·k+1 \ k⇝∞⁺} \dfrac{1}{\sqrt[j]{ | p_j | } } ≤ \varlimsup\limits_{j=2·k+1 \ k⇝∞⁺} \left | \dfrac{p_j}{p_{j+2} } \right | = \lim\limits_{k⇝∞⁺} \dfrac{ә^{2^{2 · k + 1} } }{(2 · k + 1)!} · \dfrac{(2 · k + 3)!}{ә^{2^{2 · k + 3} } } = \lim\limits_{k⇝∞⁺} \dfrac{(2 · k + 2) · (2 · k + 3)}{ә^{3·2^{2·k+1} } } ⇝ 0$ |
典例：函数$f (x) = \mathop{0}\limits_{x≤0}; \mathop{ә^{-x^{-1} } }\limits_{0<x}$在点$x_0 = 0$处连续且有无穷阶导数。其在点$x_0=0$处展开的无穷阶幂级数收敛于自身，但在点$x ≠ 0$处不收敛于自身。
| $⇓$ | $\lim\limits_{x⇝0^+} f (x) = \lim\limits_{x=t^{-1} \ t⇝∞⁺} \dfrac{1}{ә^{t} } ⇝ 0 ⇜ \lim\limits_{x⇝0^-} 0 = \lim\limits_{x⇝0^-} f (x)$ | $⇒$ | $\lim\limits_{x⇝0} f (x) = 0$ | ||
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| $⇓$ | $\dfrac{\mathrm{d}^0 f (x)}{\mathrm{d}^0 x} \mathop{==}\limits_{0<x} f (x) = ә^{-x^{-1} } · 1 = ә^{-x^{-1} } · 𝓟_0 (x^{-1})$ | $⇒$ | $\lim\limits_{x⇝0^+} \dfrac{\mathrm{d}^0 f (x)}{\mathrm{d}^0 x} ⇝ 0 ⇜ \lim\limits_{x⇝0^-} 0 = \lim\limits_{x⇝0^-} \dfrac{\mathrm{d}^0 f (x)}{\mathrm{d}^0 x}$ | ||
| $⇓$ | $\dfrac{\mathrm{d}^1 f (x)}{\mathrm{d}^1 x} \mathop{==}\limits_{0<x} \dfrac{\mathrm{d} }{\mathrm{d} x} \left[ ә^{-x^{-1} } · 𝓟_0 (x^{-1}) \right] = ә^{-x^{-1} } · \left[ \dfrac{1}{x^2} · 𝓟_0 (x^{-1}) + \dfrac{\mathrm{d} }{\mathrm{d} x} 𝓟_0 (x^{-1}) \right] = ә^{-x^{-1} } · 𝓟_2 (x^{-1})$ | $⇐$ | $\dfrac{\mathrm{d} 𝓟_0 (x^{-1})}{\mathrm{d} x} = \left.\dfrac{\mathrm{d} 𝓟_0 (y)}{\mathrm{d} y}\right | _{x^{-1} } · \dfrac{-1}{x^2} = 0$ | |
| $⇓$ | $\dfrac{\mathrm{d}^2 f (x)}{\mathrm{d}^2 x} \mathop{==}\limits_{0<x} \dfrac{\mathrm{d} }{\mathrm{d} x} \left[ ә^{-x^{-1} } · 𝓟_2 (x^{-1}) \right] = ә^{-x^{-1} } · \left[ \dfrac{1}{x^2} · 𝓟_2 (x^{-1}) + \dfrac{\mathrm{d} }{\mathrm{d} x} 𝓟_2 (x^{-1}) \right] = ә^{-x^{-1} } · 𝓟_4 (x^{-1})$ | $⇐$ | $\dfrac{\mathrm{d} 𝓟_2 (x^{-1})}{\mathrm{d} x} = \left.\dfrac{\mathrm{d} 𝓟_2 (y)}{\mathrm{d} y}\right | _{x^{-1} } · \dfrac{-1}{x^2} = 𝓟_3 (x^{-1})$ | |
| $···$ | $···$ | ||||
| $⇓$ | $\dfrac{\mathrm{d}^i f (x)}{\mathrm{d}^i x} \mathop{==}\limits_{0<x} \dfrac{\mathrm{d} }{\mathrm{d} x} \left[ ә^{-x^{-1} } · 𝓟_{2 · i - 2} (x^{-1}) \right] = ә^{-x^{-1} } · \left[ \dfrac{1}{x^2} · 𝓟_{2 · i - 2} (x^{-1}) + \dfrac{\mathrm{d} }{\mathrm{d} x} 𝓟_{2 · i - 2} (x^{-1}) \right] = ә^{-x^{-1} } · 𝓟_{2 · i} (x^{-1})$ | $⇐$ | $\dfrac{\mathrm{d} 𝓟_{2 · i - 2} (x^{-1})}{\mathrm{d} x} = \left.\dfrac{\mathrm{d} 𝓟_{2 · i - 2} (y)}{\mathrm{d} y} \right | {x^{-1} } · \dfrac{-1}{x^2} = 𝓟{2 · i - 1} (x^{-1})$ | |
| $⇓$ | $\lim\limits_{i⇝∞⁺} \lim\limits_{x⇝0^+} \dfrac{\mathrm{d}^i f (x)}{i! · \mathrm{d}^i x} = \lim\limits_{i⇝∞⁺} \lim\limits_{x=t^{-1} \ t⇝∞⁺} \dfrac{𝓟_{2 · i} (t)}{i! · ә^{t} } ⇝ 0 ⇜ \lim\limits_{x=t^{-1} \ t⇝∞⁺} \lim\limits_{i⇝∞⁺} \dfrac{𝓟_{2 · i} (t)}{i! · ә^{t} } = \lim\limits_{x⇝0^+} \lim\limits_{i⇝∞⁺} \dfrac{\mathrm{d}^i f (x)}{i! · \mathrm{d}^i x}$ | $⇒$ | \(\lim\limits_{x⇝0^+} \dfrac{\mathrm{d}^i f (x)}{i! · \mathrm{d}^i x} ⇝ 0 ⇜ \lim\limits_{x⇝0^-} 0 = \lim\limits_{x⇝0^-} \dfrac{\mathrm{d}^i f (x)}{i! · \mathrm{d}^i x}\) | ||
| $⇓$ | $f (x) = \sum\limits_{i=0}^{∞⁺} \left. \dfrac{\mathrm{d}^i f (x)}{i! · \mathrm{d}^i x} \right | {x_0=0} · x^i = 0 \mathop{≠}\limits{x∈(0, ∞⁺)} ә^{-x^{-1} }$ | $⇐$ | $\left. \dfrac{\mathrm{d}^i f (x)}{i! · \mathrm{d}^i x} \right | _{x_0=0} = 0$ |
多项式一致逼近
对于区间$[α, β]$上的函数$f (x)$可用多项式函数$P_n (x)$一致逼近，等同于多项式函数$P_n (x)$一致收敛于函数$f (x)$。
| $\left[ ∀ε>0;∃N∈ℕ;∀n≥N; \sup\limits_{x∈X} | P_n (x) - f (x) | ≤ ε \right] ⇔ [ ∀ε>0;∃N∈ℕ;∀n≥N;∀x∈X; | P_n (x) - f (x) | < ε ]$ |
| $\left[ \lim\limits_{n⇝∞⁺} P_n (x) \mathop{↭}\limits_{x∈X} f (x) \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} | P_n (x) - f (x) | \mathop{↭}\limits_{x∈X} 0 \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} \sup\limits_{x∈X} | P_n (x) - f (x) | ⇝ 0 \right]$ |
若区间$[α, β]$上的函数$f (x)$可展开成条件收敛的无穷阶幂级数，则函数$f (x)$可用多项式函数$P_n (X)$一致逼近。
$\left[ f (x) = \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n p_i · (x - x_0)^i \right] ⇒ \left[ P_n (x) ≡ \sum\limits_{i=0}^n p_i · (x - x_0)^i \right]$
若区间$(∞⁻, β]$上以及区间$[α, ∞⁺)$上的函数$f (x)$可用多项式函数$P_n (x)$一致逼近，则函数$f (x)$为多项式函数。
| $⇓$ | $\lim\limits_{n⇝∞⁺} P_n (x) \mathop{↭}\limits_{x∈[α, ∞⁺)} f (x)$ | ||||||
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| $⇓$ | $\lim\limits_{n⇝∞⁺} \sup\limits_{x∈[α, ∞⁺)} | 𝓟_{n-N} (x) | = \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝∞⁺} | 𝓟_{n-N} (x) | ⇝ 0$ | $⇐$ | $𝓟_{n-N} (x) ≡ P_n (x) - P_N (x)$ |
| $⇓$ | $𝓟_{n-N} (x) ≡ \mathrm{Con.} = 0$ | $⇐$ | $[ 𝓟_{n-N} (x) \not≡ \mathrm{Con.}] ⇒ \left[ \lim\limits_{n⇝∞⁺} \lim\limits_{x⇝∞⁺} | 𝓟_{n-N} (x) | ⇝ ∞⁺ \right]$ | ||
| $⇓$ | $f (x) = \lim\limits_{n⇝∞⁺} P_n (x) = \lim\limits_{n⇝∞⁺} [ P_N (x) + 𝓟_{n-N} (x) ] = P_N (x)$ |
若区间$[α, β]$上的函数$f (x)$可用多项式函数$P_n (x)$一致逼近，则函数$f (x)$为连续函数，反之亦然。多项式函数$P_n (x)$本身为连续函数。
反例：连续函数$f (x) = \dfrac{1}{x}$，在区间$(0^+, +1]$以及区间$[1, ∞⁺)$上不可用多项式函数$P_n (x)$一致逼近。
连续函数$f (x) = \dfrac{1}{x}$在区间$(0^+, +1]$上无确界非一致连续，但是多项式函数$P_n (x)$在区间$(0^+, +1]$上有确界且一致连续。
连续函数$f (x) = \dfrac{1}{x}$在区间$[1, ∞⁺)$上有确界且一致连续，但是多项式函数$P_n (x)$在区间$[1, ∞⁺)$上无确界且一致连续。
| 若区间$[α, β]$上的函数$f (x)$可积分，则存在多项式函数$P (x)$，使得$\int\limits_{α}^{β} | P (x) - f (x) | \mathrm{d} x = 0$。 |
| $⇓$ | $\left[ \int\limits_{α}^{β} f (x) \mathrm{d} x = I \right] ⇒ \left[ \lim\limits_{n⇝∞⁺}^{∆x_i⇝0} \sum\limits_{i=0}^n \sup\limits_{u,v∈[x_{i+0}, x_{i+1}]} | f (u) - f (v) | · ∆x_i = \lim\limits_{n⇝∞⁺}^{∆x_i⇝0} \sum\limits_{i=0}^n w_i · ∆x_i ⇝ 0 \right]$ | $⇐$ | $w_i = \sup\limits_{u,v∈[x_i, x_{i+1}]} | f (u) - f (v) | $ | ||
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| $⇓$ | $0 = \left | \begin{matrix} 1 & 1 & 1 \ x_{i} & x & x_{i+1} \ f (x_{i}) & g (x) & f (x_{i+1}) \end{matrix}\right | = \left | \begin{matrix} 1 & 0 & 0 \ x_{i} & x - x_{i} & x_{i+1} - x_{i} \ f (x_{i}) & g (x) - f (x_{i}) & f (x_{i+1}) - f (x_{i+0}) \end{matrix}\right | $ | $⇒$ | $0 = (x - x_{i}) · [f (x_{i+1}) - f (x_{i})] - (x_{i+1} - x_{i}) · [g (x) - f (x_{i})]$ | ||
| $⇓$ | $g (x) \mathop{≡≡≡≡≡}\limits_{x_i≤x≤x_{i+1} } \dfrac{x - x_{i} }{x_{i+1} - x_{i} } · [f (x_{i+1}) - f (x_{i})] + f (x_{i})$ | $⇒$ | $g (x_i) = f (x_i), g (x_{i+1}) = f (x_{i+1})$ | ||||||
| $⇓$ | $g (x) - f (x) \mathop{=====}\limits_{x_i≤x≤x_{i+1} } \dfrac{x_{i+1} - x}{x_{i+1} - x_{i} } · [ f (x_i) - f (x) ] + \dfrac{x - x_i}{x_{i+1} - x_i} · [ f (x_{i+1}) - f (x) ]$ | $⇒$ | $ | g (x) - f (x) | ≤ \dfrac{x_{i+1} - x}{x_{i+1} - x_i} · w_i + \dfrac{x - x_i}{x_{i+1} - x_i} · w_i = w_i$ | ||||
| $⇓$ | $\int\limits_{α}^{β} | g (x) - f (x) | \mathrm{d} x \mathop{====}\limits_{x_{n+1}=β}^{x_0=α} \lim\limits_{n⇝∞⁺}^{∆x_i⇝0} \sum\limits_{i=0}^n | g_n (θ_i) - f (θ_i) | · Δx_i ≤ \lim\limits_{n⇝∞⁺}^{∆x_i⇝0} \sum\limits_{i=0}^n w_i · Δx_i ⇝ 0$ | ||||
| $⇓$ | $\int\limits_{α}^{β} | P (x) - f (x) | \mathrm{d} x ≤ \int\limits_{α}^{β} | P (x) - g (x) | \mathrm{d} x + \int\limits_{α}^{β} | g (x) - f (x) | \mathrm{d} x ≤ ε_1 · (β - α) + ε_2 = ε ⇝ 0$ | $⇐$ | $\left[ P_n (x) \mathop{↭}\limits_{x∈[α, β]} g_n (x) \right] ∧ \left[ \lim\limits_{n⇝∞⁺} P_n (x) ⇝ P (x) \right]$ |
构造性多项式一致逼近
步骤一：[积分]构造性多项式函数在闭区间 $[0, 1]$上一致逼近连续函数$f (x)$，$f (x) \mathop{===}\limits_{x∉[0, 1]} f (0) = f (1) = 0$。
| $⇓$ | $\int\limits_{-1}^{+1} (1 - t^2)^n \mathrm{d} t ≥ \int\limits_{-1}^{+1} (1 - n · t^2) \mathrm{d} t = \left[ t - \dfrac{n}{3} · t^3 \right]_{-1}^{+1} = 2$ | ||||||||
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| $⇓$ | $\left[ 0 ≤ Q_n (-t) = Q_n (t) \mathop{≡}\limits_{t∈[-1, +1]} \dfrac{(1 - t^2)^n}{\int\limits_{-1}^{+1} (1 - t^2)^n \mathrm{d} t} ≤ \dfrac{(1 - t^2)^n}{2} \right] ⇒ \left[ f (x) = f (x) · \dfrac{\int\limits_{-1}^{+1} (1 - t^2)^n \mathrm{d} t}{\int\limits_{-1}^{+1} (1 - t^2)^n \mathrm{d} t} = \int\limits_{-1}^{+1} f (x) · Q_n (t) \mathrm{d} t \right]$ | ||||||||
| $⇓$ | $\left[ P_n (x) ≡ \int\limits_{-1}^{+1} f (x + t) · Q_n (t) \mathrm{d} t \mathop{===}\limits_{\mathrm{d} u = \mathrm{d} t}^{u = x + t} \int\limits_{x-1}^{x+1} f (u) · Q_n (u - x) \mathrm{d} u \mathop{=====}\limits_{-1≤x-1≤0}^{1≤x+1≤2} \int\limits_{0}^{1} f (u) · Q_n (u - x) \mathrm{d} u \right] ⇒ \left[ | P_n (x) - f (x) | ≤ \int\limits_{-1}^{+1} | f (x + t) - f (x) | · | Q_n (t) | \mathrm{d} t \right]$ | ||
| $⇓$ | $[ ∀ε>0;∃δ>0; | t | = | (x + t) - x | < δ \mathop{⇒}\limits_{x,y∈[-1, +1]} | f (x + t) - f (x) | < ε ] ∧ \left[ | f (x) - f (y) | \mathop{≤}\limits_{x,y∈[0, 1]} \mathrm{Sup.} ≡ \sup\limits_{x∈[0, 1]} f (x) - \inf\limits_{x∈[0, 1]} f (x) \right]$ |
| $⇓$ | $\lim\limits_{n⇝∞⁺} | P_n (x) - f (x) | ≤ \lim\limits_{n⇝∞⁺} \left[ 2 · \mathrm{Sup.} · \int\limits_{+δ}^{+1} Q_n (t) \mathrm{d} t + 2 · ε · \int\limits_{0}^{+δ} Q_n (t) \mathrm{d} t \right] ≤ \lim\limits_{n⇝∞⁺} \left[ 2 · \mathrm{Sup.} · \dfrac{(1 - δ^2)^n}{2} · 1 + 2 · ε · \dfrac{(1 - 0^2)^n}{2} · 1 \right] \mathop{⇝}\limits^{ε⇝0^+} 0$ | ||||||
| $⇓$ | $\lim\limits_{n⇝∞⁺} P_n (x) \mathop{↭}\limits_{x∈[0， 1]} f (x)$ |
步骤二：[积分]构造性多项式函数在闭区间$[0, 1]$上一致逼近连续函数$g (x)$，$f (x) \mathop{=}\limits_{x∈[0, 1]} x · [ g (1) - g (0) ] - [ g (x) - g (0) ]$，$f (x) \mathop{=}\limits_{x∉[0, 1]} f (0) = f (1) = 0$。
$\left[ \lim\limits_{n⇝∞⁺} P_n (x) \mathop{↭}\limits_{x∈[0, 1]} f (x) = x · [ g (1) - g (0) ] - [ g (x) - g (0) ] \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} x · [ g (1) - g (0)] + g (0) -P_n (x) \mathop{↭}\limits_{x∈[0, 1]} g (x) ] \right]$
步骤三：[积分]构造性多项式函数在闭区间$[α, β]$上一致逼近连续函数$h (y)$，$y = α + x · (β - α), x = \dfrac{y - α}{β - α}$。
$\left[ \lim\limits_{n⇝∞⁺} P_n (x) \mathop{↭}\limits_{x∈[0, 1]} g (x) \mathop{=}\limits_{x∈[0, 1]} h (y) \mathop{=}\limits_{y∈[α, β]} h [α + x · (β - α)] \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} P_n \left( \dfrac{y - α}{β - α} \right) \mathop{↭}\limits_{x∈[0, 1]} h (y) \right]$
附加证明：$(1 - t^2)^n \mathop{≥}\limits_{-1≤t≤1} (1 - n · t^2)$。
| $⇓$ | $g (x) ≡ (1 - t^2)^n - (1 - n · t^2)$ | ||
|---|---|---|---|
| $⇓$ | $\dfrac{\mathrm{d} }{\mathrm{d} x} g (x) = n · (1 - t^2)^{n - 1} · (-2 · t) + 2 · n · t = 2 · n · t · [ 1 - (1 - t^2)^{n - 1} ]$ | $⇒$ | $\dfrac{\mathrm{d} }{\mathrm{d} x} g (x) ≈ \mathop{<0}\limits_{-1≤x<0}; \mathop{=0}\limits_{x=0};\mathop{>0}\limits_{0<x≤+1};$ |
| $⇓$ | $g (x) \mathop{≥}\limits_{-1≤x≤+1} g (0) = 0$ | $⇒$ | $(1 - t^2)^n \mathop{≥}\limits_{-1≤x≤+1} (1 - n · t^2)$ |
步骤一：[离散]构造性多项式函数在闭区间$[0, 1]$上一致逼近连续函数$f (x)$，$f (x) \mathop{===}\limits_{x∉[0, 1]} 0$。
| $⇓$ | $P_n [f (t) = 1; x; y ]{y = 1 - x} ≡ \sum\limits{i=0}^n \dfrac{n!}{i! · (n - i)!} · x^i · y^{n - i} = (x + y)^n = 1$ | ||||||
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| $⇓$ | $\sum\limits_{i=1}^n \dfrac{n!}{(i - 1)! · (n - i)!} · x^{i - 1} · y^{n - i} = \dfrac{∂}{∂ x} (x + y)^n = n · (x + y)^{n - 1}$ | ||||||
| $⇓$ | $P_n [f (t) = t; x; y ]{y=1-x} ≡ \sum\limits{i=0}^{n} \dfrac{i}{n} · \dfrac{n!}{i! · (n - i)!} · x^i · y^{n - i} = x · (x + y)^{n - 1} = x$ | ||||||
| $⇓$ | $\sum\limits_{i=1}^n \dfrac{i}{n} · \dfrac{n!}{(i - 1)! · (n - i)!} · x^{i - 1} · y^{n - i} = \dfrac{∂}{∂ x} [x · (x + y)^{n - 1}] = (x + y)^{n - 1} + (n - 1) · x · (x + y)^{n - 2}$ | ||||||
| $⇓$ | $P_n [f (t) = t^2; x; y ]{y=1-x} ≡ \sum\limits{i=0}^n \dfrac{i^2}{n^2} · \dfrac{n!}{i! · (n - i)!} · x^{i} · y^{n - 1} = \dfrac{x}{n} · [ (x + y)^{n - 1} + (n - 1) · x · (x + y)^{n - 2} ] = \left( 1 - \dfrac{1}{n} \right) · x^2 + \dfrac{1}{n} · x$ | ||||||
| $⇓$ | $\sum\limits_{i=0}^n \left[ \dfrac{i}{n} - x \right]^2 · \dfrac{n!}{i! · (n - i)!} · x^i · (1 - x)^{n - i} = \sum\limits_{i=0}^n \left[ \dfrac{i^2}{n^2} - 2 · x · \dfrac{i}{n} + x^2 \right] · \dfrac{n!}{i! · (n - i)!} · x^i · (1 - x)^{n - i}$ | ||||||
| $⇓$ | $\sum\limits_{i=0}^n \left[ \dfrac{i}{n} - x \right]^2 · \dfrac{n!}{i! · (n - i)!} · x^i · (1 - x)^{n - i} = \left[ \left( 1 - \dfrac{1}{n} \right) · x^2 + \dfrac{1}{n} · x \right] - 2 · x · x + x^2 · 1 = \dfrac{1}{n} · x · (1 - x) ≤ \dfrac{1}{n} · \dfrac{1}{2} · \left( 1 - \dfrac{1}{2} \right) = \dfrac{1}{4 · n}$ | ||||||
| $⇓$ | $[ ∀ε>0;∃δ>0; | y - x | < δ \mathop{⇒}\limits_{x,y∈[0, 1]} | f (y) - f (x) | < ε ] ∧ \left[ | f (y) - f (x) | \mathop{≤}\limits_{x,y∈[0, 1]} \mathrm{Sup.} ≡ \sup\limits_{x∈[0, 1]} f (x) - \inf\limits_{x∈[0, 1]} f (x) \right] ∧ \left[ f (x) = \sum\limits_{i=0}^n f (x) · \dfrac{n!}{i! · (n - i)!} · x^i · (1 - x)^{n - i} \right]$ |
| $⇓$ | $\sum\limits_{i=0}^n \left | f \left( \dfrac{i}{n} \right) - f (x) \right | · \dfrac{n!}{i! · (n - i)!} · x^i · (1 - x)^{n - i} ≤ \left[ ε · \sum\limits_{\left | \frac{i}{n} - x \right | <δ}^{n} \dfrac{n!}{i! · (n - i)!} · x^i · (1 - x)^{n - i} + \dfrac{\mathrm{2 · Sup.} }{δ^2} · \sum\limits_{\left | \frac{i}{n} - x \right | ≥δ}^n \left[ \dfrac{i}{n} - x \right]^2 · \dfrac{n!}{i! · (n - i)!} · x^i · (1 - x)^{n - i} \right]$ |
| $⇓$ | $\lim\limits_{n⇝∞⁺} | P_n [f (t); x; y ]_{y=1-x} - f (x) | ≤ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n \left | f \left( \dfrac{i}{n} \right) - f (x) \right | · \dfrac{n!}{i! · (n - i)!} · x^i · (1 - x)^{n - i} ≤ \lim\limits_{n⇝∞⁺} \left[ ε · 1 + \dfrac{\mathrm{2 · Sup.} }{δ^2 · 4 · n} \right] \mathop{⇝}\limits^{ε⇝0^+} 0$ | ||
| $⇓$ | $\lim\limits_{n⇝∞⁺} P_n [f (t); x; y ]{y=1-x} \mathop{↭}\limits{x∈[0, 1]} f (x)$ |
步骤二：[离散]构造性多项式函数在闭区间$[α, β]$上一致逼近连续函数$h (y)$，$y = α + x · (β - α), x = \dfrac{y - α}{β - α}$。
$\left[ \lim\limits_{n⇝∞⁺} P_n (x) \mathop{↭}\limits_{x∈[0, 1]} f (x) \mathop{=}\limits_{x∈[0, 1]} h (y) \mathop{=}\limits_{y∈[α, β]} h [α + x · (β - α)] \right] ⇔ \left[ \lim\limits_{n⇝∞⁺} P_n \left( \dfrac{y - α}{β - α} \right) \mathop{↭}\limits_{x∈[0, 1]} h (y) \right]$
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