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 ?

Hint. One may use a Taylor series expansion, as $n \to \infty$,
$$
\arctan \frac1n=\frac1n+O\left(\frac1{n^3}\right)
$$  giving, as $n \to \infty$,
$$
\left(\frac1n-\arctan \frac1n\right)^a=O\left(\frac1{n^{3a}}\right)
$$ I hope you can take it from here.
