证明无穷级数求和公式 ∑(k=1,∞)1／k^2=π^2／6

wintex

002 請問數學

13668164639

答案如下答案如下

luyuanhong

下面是我过去在《数学中国》发表过的一个帖子：

王守恩

一种另类解法，希望对你有启发。

\(\ \ 1=1×1=1\)

\(\ \displaystyle\frac{1}{3}=3×\frac{1}{3^2}<\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}\)

\(\displaystyle\frac{1}{3^2}=9×\frac{1}{9^2}<\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+...+\frac{1}{13^2}\)

\(\displaystyle\frac{1}{3^3}=27×\frac{1}{27^2}<\frac{1}{14^2}+\frac{1}{15^2}+\frac{1}{16^2}+...+\frac{1}{40^2}\)

\(\displaystyle\frac{1}{3^4}=81×\frac{1}{81^2}<\frac{1}{41^2}+\frac{1}{42^2}+\frac{1}{43^2}+...+\frac{1}{121^2}\)

\(\ ......\)

\(\displaystyle左边=1+\frac{1}{3}\ +\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+\frac{1}{3^5}+......=\frac{3}{3}×\frac{3}{2}=\frac{3^2}{6}\)

\(\displaystyle右边=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+......=\frac{\pi}{3}×\frac{\pi}{2}=\frac{\pi^2}{6}\)