# 高斯判别法

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

## 定理

${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ 是要判断审敛性的级数，其中（至少从某一项开始）${\displaystyle a_{n}>0}$ 。倘若其相邻项比值${\displaystyle {\frac {a_{n}}{a_{n+1}}}}$ 可以被表示为：

${\displaystyle {\frac {a_{n}}{a_{n+1}}}=\lambda +{\frac {\mu }{n}}+{\frac {\theta _{n}}{n^{2}}}}$

• ${\displaystyle \lambda >1}$ ${\displaystyle \lambda =1,\mu >1}$ 时，级数收敛；
• ${\displaystyle \lambda <1}$ ${\displaystyle \lambda =1,\mu \leq 1}$ 时，级数发散。

## 参考文献

1. ^ Konrad Knopp. Theory and Application of Infinite Series. London: Blackie & Son Ltd. 1954.
2. ^ Sayel A. Ali. The mth Ratio Test: New Convergence Test for Series. The American Mathematical Monthly. 2008, 115 (6): 514–524.
3. ^ Kyle Blackburn. The mth Ratio Convergence Test and Other Unconventional Convergence Tests (PDF). University of Washington College of Arts and Sciences. 4 May 2012 [27 November 2018].
4. ^ František Ďuriš. Infinite series: Convergence tests (Bachelor's thesis). Katedra Informatiky, Fakulta Matematiky, Fyziky a Informatiky, Univerzita Komenského, Bratislava. 2009 [28 November 2018].
5. ^ Г. М. 菲赫金哥尔茨. 微积分学教程（第二卷）（第8版） 第二版. 2006: 230. ISBN 978-7-04-018304-7.