TITLE: Does the series $\;\sum\limits_{n=0}^{\infty}\left(\frac{\pi}{2} - \arctan(n)\right)$ converge or diverge?
QUESTION [14 upvotes]: I tried everything that I know but I couldn't solve this series:
$$\sum_{n=0}^{\infty} \left(\frac{\pi}{2} - \arctan(n)\right)$$
Does it diverge or converge?

REPLY [6 votes]: The serie $\sum \arctan \frac{1}{n}$ is divergent for two reasons: first, it's a positive serie and second $\arctan\frac{1}{n}\sim \frac{1}{n}$ so by comparaison we conclude since the serie $\sum \frac{1}{n}$ is divergent.
To explain more why the positivity of the serie is important to use comparaison test, this is a counterexample : the serie $\sum \log(1+\frac{(-1)^n}{\sqrt{n}})$ is divergent while $\log(1+\frac{(-1)^n}{\sqrt{n}})\sim\frac{(-1)^n}{\sqrt{n}}$ and the serie $\sum \frac{(-1)^n}{\sqrt{n}}$ is convergent