problem on convergence of series $\sum_{n=1}^\infty\left(\frac1n-\tan^{-1} \frac1n\right)^a$ Finding the set of all positive values of $a$ for which the series
$$
\sum_{n=1}^\infty\left(\frac1n-\tan^{-1} \frac1n\right)^a
$$ converges.
How will it depend on the the value of a that is its power of the term?
After expanding the arc tan term I get the form of summation of $[ -1/n^3(1/3+1/n^2+......]^a $. now how does it depend on a ?

Since you don't ask for which values of $a$ the series converges, here is an heuristic answer.
When $n$ goes to $\infty$, $\frac 1n$ goes to $0$ so $\tan^{—1}(1/n)$ goes to $0$.
So the general term (without the "$a$") of your series goes to $0$, so there is a reasonable chance that the series will converge.
It will converge if the general term goes to $0$ sufficiently fast.
That would be the case if $a$ is sufficiently big, but it won't be the case if $a$ is too small.
