# Is the series $\sum (\frac{1}{2} - \frac{1}{\pi} \tan^{-1}(n) )$ convergent$?$

Is the series $$\sum (\frac{1}{2} - \frac{1}{\pi} \tan^{-1}(n) )$$ convergent$$?$$

I couldn't find any suitable test for this series. I tried tests like comparison,root test, ratio test, raabe's, gauss, nothing is working.

Any suggestion$$?$$

Use the fact that $$\arctan\left(\frac{1}{n}\right)+\arctan(n)=\frac{\pi}{2},$$
for all $$n>0$$ and that $$\arctan\left(\frac{1}{n}\right)\sim\frac{1}{n}\quad \text{when }n\to\infty ,$$ to prove that it diverges.
• Oh I confused $\tan^{-1}$ with $\tanh$. – orlp Sep 22 '19 at 14:53