조화수열과 조화급수

개요

\[\gamma=0.577215664901532860606512090\cdots\]

오일러-맥클로린 공식 을 통해 다음을 얻는다 \[H_{n}=\sum_{k=1}^{n}\frac{1}{k}\sim \log n +\gamma+\frac{1}{2n}-\sum_{s=1}^{\infty}\frac{B_{2s}}{(2s)n^{2s}}\]
다음이 성립한다 $$H_{n}= \log n +\gamma+ O(1/n)$$

$H_{n-1}=H_n-\frac{1}{n}$

$H_ {n-1}^2=(H_n-\frac{1}{n})^2=H_n^2+\frac{1}{n^2}-\frac{2H_n}{n}$

$\sum_{n=1}^\infty H_nz^n = \frac {-\ln(1-z)}{1-z}$

$\sum_{n=1}^\infty \frac{H_n}{n+1}z^{n+1} =\frac{1}{2}\log^2(1-z)$

$\sum_{n=1}^\infty \frac{H_n}{n}z^n =\operatorname{Li}_ 2(z)+\frac{1}{2}\log^2(1-z)$

$z=e^{it}$, $0 \leq t \leq \pi$ 에서

위 식의 실수부를 취하면, 각각 다음 식을 얻는다.

$\sum_{n=1}^\infty \frac{H_n}{n+1}\sin (n+1)t=\frac{1}{2}(t-\pi)\log(2\sin\frac{t}{2})$

$\sum_{n=1}^\infty \frac{H_n}{n}\sin nt=\operatorname{Cl}_ 2(t)+\frac{1}{2}(t-\pi)\log(2\sin\frac{t}{2})$

$\sum_{n=1}^{\infty}\frac{H_n^2}{(n+1)^2}=\frac{11\pi^4}{360}$

$\sum_{n=1}^{\infty}\frac{H_n^2}{n^2}=\frac{17\pi^4}{360}$

$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}$