# 幂级数

${\displaystyle \zeta (s)=\sum _{k=1}^{\infty }{\frac {1}{k^{s}}}}$

${\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-c\right)^{n}}$
${\displaystyle =a_{0}+a_{1}(x-c)^{1}+a_{2}(x-c)^{2}+a_{3}(x-c)^{3}+\cdots }$

## 例子

${\displaystyle f(x)=3+2x+1x^{2}+0x^{3}+0x^{4}+\cdots }$

${\displaystyle f(x)=6+4(x-1)+1(x-1)^{2}+0(x-1)^{3}+0(x-1)^{4}+\cdots }$

${\displaystyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}=1+x+x^{2}+x^{3}+\cdots }$，是幂级数中基本而又重要的一类。同样重要的还有指数的幂级数展开：
${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots ,}$

${\displaystyle \sin(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots ,}$

## 敛散性

${\displaystyle \sum _{n=0}^{\infty }|a_{n}\,x^{n}|=\sum _{n=1}^{\infty }\left(|a_{n}|\,r_{0}^{n}\right)\cdot \left({\frac {|x|}{r_{0}}}\right)^{n}}$
${\displaystyle \ \ \leq \sum _{n=0}^{\infty }M\cdot \left({\frac {|x|}{r_{0}}}\right)^{n}}$
${\displaystyle \ \ =M\cdot \sum _{n=0}^{\infty }\left({\frac {|x|}{r_{0}}}\right)^{n}}$

1. 要么对所有的非零复数，${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}}$都发散；
2. 要么存在一个正常数（包括正无穷）${\displaystyle R}$，使得当${\displaystyle |x|时，${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}}$绝对收敛，当${\displaystyle |x|>R}$时，${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}}$发散。

 ${\displaystyle \rho }$是正实数时，${\displaystyle R={1 \over \rho }}$。 ${\displaystyle \rho =0}$时，${\displaystyle R=\infty }$。 ${\displaystyle \rho =\infty }$时，${\displaystyle R=0}$。

${\displaystyle R=\liminf _{n\to \infty }\left|a_{n}\right|^{-{\frac {1}{n}}}}$

## 幂级数的运算

${\displaystyle (a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n}+\cdots )\pm (b_{0}+b_{1}x+b_{2}x^{2}+\cdots +b_{n}x^{n}+\cdots )=(a_{0}+b_{0})+(a_{1}+b_{1})x+(a_{2}+b_{2})x^{2}+\cdots +(a_{n}+b_{n})x^{n}+\cdots }$

${\displaystyle \left(\sum _{n=0}^{\infty }a_{n}(x-c)^{n}\right)\left(\sum _{n=0}^{\infty }b_{n}(x-c)^{n}\right)}$
${\displaystyle =\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }a_{i}b_{j}(x-c)^{i+j}}$
${\displaystyle =\sum _{n=0}^{\infty }\left(\sum _{i=0}^{n}a_{i}b_{n-i}\right)(x-c)^{n}}$

## 一致收敛性

${\displaystyle f:(-R,R)\longrightarrow \mathbb {R} }$
${\displaystyle .\ \ \ \ \ \ x\longmapsto \sum _{n=0}^{\infty }a_{n}x^{n}}$

## 幂级数函数的求导和积分

${\displaystyle f^{\prime }(x)=\sum _{n=1}^{\infty }a_{n}nx^{n-1}=\sum _{n=0}^{\infty }a_{n+1}\left(n+1\right)x^{n}}$
${\displaystyle \int f(x)\,dx=\sum _{n=0}^{\infty }{\frac {a_{n}x^{n+1}}{n+1}}+k=\sum _{n=1}^{\infty }{\frac {a_{n-1}x^{n}}{n}}+k}$

## 函数的幂级数展开

${\displaystyle \forall z\in D(c,R),\qquad f(z)=\sum _{n=0}^{+{\infty }}a_{n}(z-c)^{n}}$

${\displaystyle \forall n\in \mathbb {N} ,\,a_{n}={f^{(n)}(c) \over {n!}}}$

### 函数的可展性

x>0时，${\displaystyle f(x)=e^{-1/x^{2}}}$
${\displaystyle x\leq 0}$时，${\displaystyle f(x)=0}$

${\displaystyle |f^{n}(x)|\leq M^{n}}$，那么${\displaystyle f}$可以在c附近展开成幂级数：
${\displaystyle \forall x\in (c-r,c+r),\ \ f(x)=\sum _{n=0}^{+{\infty }}{f^{(n)}(c) \over {n!}}(x-c)^{n}}$

### 常见函数的幂级数展开

1. ${\displaystyle \forall x\in \mathbb {C} ,\,e^{x}=\sum _{n=0}^{+{\infty }}{\frac {x^{n}}{n!}}.}$

2. ${\displaystyle \forall x\in \mathbb {R} ,\,\cos x=\sum _{n=0}^{+{\infty }}(-1)^{n}\,{\frac {x^{2\,n}}{(2\,n)!}}.}$

3. ${\displaystyle \forall x\in \mathbb {R} ,\,\sin x=\sum _{n=0}^{+{\infty }}(-1)^{n}\,{\frac {x^{2\,n+1}}{(2\,n+1)!}}.}$

4. ${\displaystyle \forall x\in \mathbb {R} ,\,\operatorname {ch} \,x=\sum _{n=0}^{+{\infty }}{\frac {x^{2\,n}}{(2\,n)!}}.}$

5. ${\displaystyle \forall x\in \mathbb {R} ,\,\operatorname {sh} \,x=\sum _{n=0}^{+{\infty }}{\frac {x^{2\,n+1}}{(2\,n+1)!}}.}$

6. ${\displaystyle \forall x\in D(0,1),\,{1 \over {1-x}}=\sum _{n=0}^{+{\infty }}{x^{n}}.}$

7. ${\displaystyle \forall x\in (-1,1],\,\ln(1+x)=\sum _{n=1}^{+{\infty }}(-1)^{n+1}{x^{n} \over {n}}.}$

8. ${\displaystyle \forall x\in [-1,1],\,\arctan \,x=\sum _{n=0}^{+{\infty }}(-1)^{n}\,{\frac {x^{2\,n+1}}{2\,n+1}}\;}$，特别地，${\displaystyle \pi =4\,\sum _{n=0}^{+{\infty }}{\frac {(-1)^{n}}{2\,n+1}}}$

9. ${\displaystyle \forall x\in \,(-1,1),\ \forall \alpha \,\not \in \,\mathbb {N} ,\,(1+x)^{\alpha }\,=1\;+\;\sum _{n=1}^{+{\infty }}{{\frac {\alpha \,(\alpha -1)\cdots (\alpha -n+1)}{n!}}\,x^{n}}.}$

10. ${\displaystyle \forall x\in \mathbb {R} ,\,\forall \alpha \,\in \,\mathbb {N} ,\,(1+x)^{\alpha }\,=1\;+\;\sum _{n=1}^{+{\infty }}{{\frac {\alpha \,(\alpha -1)\cdots (\alpha -n+1)}{n!}}\,x^{n}}=\sum _{n=0}^{\alpha }{{\alpha \choose n}\,x^{n}}.}$

11. ${\displaystyle \forall x\in (-1,1),\,\operatorname {artanh} \,x=\sum _{n=0}^{+{\infty }}\,{\frac {x^{2\,n+1}}{2\,n+1}}.}$

12. ${\displaystyle \forall x\in (-1,1),\,\arcsin \,x=x+\sum _{n=1}^{+{\infty }}\,\left({\frac {\prod _{k=1}^{n}\,(2\,k-1)}{\prod _{k=1}^{n}\,2\,k}}\right){\frac {x^{2\,n+1}}{2\,n+1}}}$

13. ${\displaystyle \forall x\in (-1,1),\,\operatorname {arsinh} \,x=x+\sum _{n=0}^{+{\infty }}\,(-1)^{n}\,\left({\frac {\prod _{k=1}^{n}\,(2\,k-1)}{\prod _{k=1}^{n}\,2\,k}}\right){\frac {x^{2\,n+1}}{2\,n+1}}}$

14. ${\displaystyle \forall x\in \,\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right),\ \tan x={\frac {2}{\pi }}\,\sum _{n=0}^{+{\infty }}\,{\left({\frac {x}{\pi }}\right)}^{2\,n+1}(2^{2\,n+2}-1)\;\zeta (2\,n+2)}$，其中${\displaystyle \forall p>1,\,\zeta (p)=\sum _{n=1}^{+{\infty }}\,{\frac {1}{n^{p}}}}$

## 多元幂级数

${\displaystyle f(x_{1},\dots ,x_{n})=\sum _{j_{1},\dots ,j_{n}=0}^{\infty }a_{j_{1},\dots ,j_{n}}\prod _{k=1}^{n}\left(x_{k}-c_{k}\right)^{j_{k}},}$

${\displaystyle f(x)=\sum _{\alpha \in \mathbb {N} ^{n}}a_{\alpha }\left(x-c\right)^{\alpha }.}$

## 参考来源

1. ^ 史济怀，组合恒等式，中国科学技术大学出版社，2001