Post History
#2: Post edited
by
Snoopy
·
2024-01-20T13:28:05Z (10 months ago)
$ \left|\sum_{n=1}^{\infty}\frac{m}{n^2+m^2} ight|<\frac{\pi}{2} $
- For any real number $m$, $ \left|\sum_{n=1}^{\infty}\frac{m}{n^2+m^2} ight|<\frac{\pi}{2} $
#1: Initial revision
by
Snoopy
·
2024-01-20T13:25:32Z (10 months ago)
$ \left|\sum_{n=1}^{\infty}\frac{m}{n^2+m^2}\right|<\frac{\pi}{2} $
> **Problem.** Prove that for any real number $m$,
$$
\left|\sum_{n=1}^{\infty}\frac{m}{n^2+m^2}\right|<\frac{\pi}{2}
$$
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**Notes.** This is an exercise in calculus. There are different ways to approach this problem. One may find the sum on the left explicitly with hyperbolic functions. One can also solve this problem by elementary estimates of integrals; I will write this one below.