# 阿贝尔判别法

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

## 实数项级数的阿贝尔判别法

• ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$收敛
• ${\displaystyle \lbrace b_{n}\rbrace \,}$单调的，${\displaystyle \lim _{n\rightarrow \infty }b_{n}\neq \infty }$

${\displaystyle \sum _{n=1}^{\infty }a_{n}b_{n}}$

## 复数项级数的阿贝尔判别法

${\displaystyle \lim _{n\rightarrow \infty }a_{n}=0\,}$

${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}\,}$

## 证明

${\displaystyle z=e^{i\theta }\quad \Rightarrow \quad z^{\frac {1}{2}}-z^{-{\frac {1}{2}}}=2i\sin {\textstyle {\frac {\theta }{2}}}\neq 0}$

{\displaystyle {\begin{aligned}2i\sin {\textstyle {\frac {\theta }{2}}}\left(S_{p}-S_{q}\right)&=\sum _{n=q+1}^{p}a_{n}\left(z^{n+{\frac {1}{2}}}-z^{n-{\frac {1}{2}}}\right)\\&=\left[\sum _{n=q+2}^{p}\left(a_{n-1}-a_{n}\right)z^{n-{\frac {1}{2}}}\right]-a_{q+1}z^{q+{\frac {1}{2}}}+a_{p}z^{p+{\frac {1}{2}}}\,\end{aligned}}}

${\displaystyle S_{p}=\sum _{n=0}^{p}a_{n}z^{n}.\,}$

{\displaystyle {\begin{aligned}\left|2i\sin {\textstyle {\frac {\theta }{2}}}\left(S_{p}-S_{q}\right)\right|&=\left|\sum _{n=q+1}^{p}a_{n}\left(z^{n+{\frac {1}{2}}}-z^{n-{\frac {1}{2}}}\right)\right|\\&\leq \left[\sum _{n=q+2}^{p}\left|\left(a_{n-1}-a_{n}\right)z^{n-{\frac {1}{2}}}\right|\right]+\left|a_{q+1}z^{q+{\frac {1}{2}}}\right|+\left|a_{p}z^{p+{\frac {1}{2}}}\right|\\&=\left[\sum _{n=q+2}^{p}\left(a_{n-1}-a_{n}\right)\right]+a_{q+1}+a_{p}\\&=a_{q+1}-a_{p}+a_{q+1}+a_{p}=2a_{q+1}\,\end{aligned}}}

## 注解

1. ^ (Moretti, 1964, p. 91)

## 参考文献

• Gino Moretti, Functions of a Complex Variable, Prentice-Hall, Inc., 1964