# 阿贝尔判别法

$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{k^s}$

## 实数项级数的阿贝尔判别法

• $\sum^{\infty}_{n=1}a_n$收敛
• $\lbrace b_n \rbrace\,$单调的，$\lim_{n \rightarrow \infty} b_n \ne \infty$

$\sum^{\infty}_{n=1}a_n b_n$

## 复数项级数的阿贝尔判别法

$\lim_{n\rightarrow\infty} a_n = 0\,$

$f(z) = \sum_{n=0}^\infty a_nz^n\,$

## 证明

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

\begin{align} 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_pz^{p+\frac{1}{2}}\, \end{align}

$S_p = \sum_{n=0}^p a_nz^n.\,$

\begin{align} \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| \\ & \le \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_pz^{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{align}

## 注解

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

## 参考文献

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