調和級数

harmonic series

正の整数の逆数の和

$ \sum_{n=1}^\infty \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} \cdots

調和級数は発散する

code:tex

\begin{aligned}

I

&= 1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right) + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right) + \cdots\\

&> 1 + \frac{1}{2} + \left(\frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) + \cdots\\

J

&= 1 + \frac{1}{2} + \frac{1}{2} +\cdots\\

&= \sum_{k=1}^\infty \left(1 + \frac{1}{2}k\right)

\end{aligned}