Compute $\sum_{n=1}^\infty \frac{1}{n\pi}$

Question

Question :

Compute $\sum_{n=1}^\infty \frac{1}{n\pi}$

Wolfram Alpha says $\sum_{n=1}^m\frac1{n\pi}=\frac{H_m}{π}$ where $H_m$ equals $m^{th}$ harmonic number.

Can anybody help me derive it and explain why it is so? Being a high-schooler, please don't think I am dumb!

Answer 1

Note that, for each $N\in\Bbb N$,$$\sum_{n=1}^N\frac1{\pi n}=\frac1\pi\sum_{n=1}^N\frac1n.$$But$$\lim_{N\to\infty}\sum_{n=1}^N\frac1n=\infty;$$in other words, the harmonic series diverges. So, your sum doesn't exist (in $\Bbb R$).