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.

  • $\begingroup$ @orlp: it's quite famous and very easy to prove. $\endgroup$ – Surb Sep 22 '19 at 14:52
  • $\begingroup$ Oh I confused $\tan^{-1}$ with $\tanh$. $\endgroup$ – orlp Sep 22 '19 at 14:53
  • $\begingroup$ $Surb, I didn't expect, the solution could be that easy. I was trying to apply test directly on the given series and I forgot about simple equations. $\endgroup$ – Mathsaddict Sep 22 '19 at 14:58
  • $\begingroup$ @Mathsaddict: Don't worry, that happen sometimes :-) $\endgroup$ – Surb Sep 22 '19 at 15:05

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