Q:

The set of all positive values of $a$ for which the series $\sum_{n=1}^\infty ((1/n) - \tan^{-1}(1/n))^a$ converges

The set of all positive values of $a$ for which the series
$$\displaystyle\sum_{n=1}^\infty ((1/n) - \tan^{-1}(1/n))^a$$
converges, is
(A) $(0,1/3]$
(B) $(0,1/3)$
(C) $[1/3,\infty)$
(D) $(1/3,\infty)$

How can I solve this problem?  I am totally helpless.  Thanks for your help  . 

A:

$$\arctan \frac{1}{x}=\frac{1}{x}-\frac{1}{3x^3}+\frac{1}{5x^5}-\cdots $$
Which yields $$\frac{1}{n}-\arctan \frac{1}{n}=\frac{1}{3n^3}-\frac{1}{5n^5}+\frac{1}{7n^7}-\cdots$$
Hence deduce that: $$\left(\frac{1}{3n^3}-\frac{1}{5n^5}\right)^a<\left(\frac{1}{n}-\arctan \frac{1}{n}\right)^a<\left(\frac{1}{3n^{3}}\right)^a$$
From this it is clear that we only have convergence for $a>\frac{1}{3}$

