有名な級数をまとめただけの記事です。
目次
ゼータ関数
$$\zeta(s)=\displaystyle\sum_{k=1}^{\infty}\displaystyle\frac{1}{k^s}$$
\(s=2\) バーゼル問題
$$\zeta(2)=\displaystyle\sum_{k=1}^{\infty}\displaystyle\frac{1}{k^2}=\displaystyle\frac{1}{1^2}+\displaystyle\frac{1}{2^2}+\displaystyle\frac{1}{3^2}+\cdots =\displaystyle\frac{\pi^2}{6}$$
\(s=4\)
$$\zeta(4)=\displaystyle\sum_{k=1}^{\infty}\displaystyle\frac{1}{k^4}=\displaystyle\frac{1}{1^4}+\displaystyle\frac{1}{2^4}+\displaystyle\frac{1}{3^4}+\cdots =\displaystyle\frac{\pi^4}{90}$$
\(s=1\)
$$\zeta(1)=\displaystyle\sum_{k=1}^{\infty}\displaystyle\frac{1}{k}=\displaystyle\frac{1}{1}+\displaystyle\frac{1}{2}+\displaystyle\frac{1}{3}+\cdots =\infty$$
\(s=-1\) 解析接続
$$\zeta(-1)=\displaystyle\sum_{k=1}^{\infty}\displaystyle\frac{1}{k^{-1}}=1+2+3+4+\cdots =-\displaystyle\frac{1}{12}$$
フーリエ級数関連
$$\displaystyle\sum_{k=1}^{\infty}\displaystyle\frac{1}{(2k-1)^2}=\displaystyle\frac{1}{1^1}+\displaystyle\frac{1}{3^2}+\displaystyle\frac{1}{5^2}+\cdots =\displaystyle\frac{\pi^2}{8}$$
$$\displaystyle\sum_{k=1}^{\infty}\displaystyle\frac{(-1)^{k-1}}{k^2}=\displaystyle\frac{1}{1^2}-\displaystyle\frac{1}{2^2}+\displaystyle\frac{1}{3^2}-\displaystyle\frac{1}{4^2}+\cdots =\displaystyle\frac{\pi^2}{12}$$
有名級数
メルカトル級数
$$1-\displaystyle\frac{1}{2}+\displaystyle\frac{1}{3}-\displaystyle\frac{1}{4}+\cdots=\displaystyle\sum_{k=1}^{\infty}\displaystyle\frac{(-1)^{k-1}}{k}=\log 2$$
ライプニッツ級数
$$1-\displaystyle\frac{1}{3}+\displaystyle\frac{1}{5}-\displaystyle\frac{1}{7}\cdots=\displaystyle\sum_{k=1}^{\infty}\displaystyle\frac{(-1)^{k-1}}{2k-1}=\displaystyle\frac{\pi}{4}$$
