Answer
The series converges.
Work Step by Step
Notice that the series has positive terms.
Write ${a_k} = f\left( k \right)$, where $f\left( x \right) = \dfrac{{{{\tan }^{ - 1}}x}}{{1 + {x^2}}}$.
From the figure attached, we see that $f$ is continuous and decreasing for $x \ge 1$. Thus, we can use the integral test to check the convergence of the series.
Evaluate $\mathop \smallint \limits_1^\infty \dfrac{{{{\tan }^{ - 1}}x}}{{1 + {x^2}}}{\rm{d}}x$.
Let $t = {\tan ^{ - 1}}x$. So, $dt = \dfrac{1}{{1 + {x^2}}}dx$. The integral becomes
$\mathop \smallint \limits_1^\infty \dfrac{{{{\tan }^{ - 1}}x}}{{1 + {x^2}}}{\rm{d}}x = \mathop \smallint \limits_{\pi /4}^{\pi /2} t{\rm{d}}t = \dfrac{1}{2}\left( {{t^2}|_{\pi /4}^{\pi /2}} \right) = \dfrac{1}{2}\left( {\dfrac{{{\pi ^2}}}{4} - \dfrac{{{\pi ^2}}}{{16}}} \right) = \dfrac{{3{\pi ^2}}}{{32}}$
Since $\mathop \smallint \limits_1^\infty \dfrac{{{{\tan }^{ - 1}}x}}{{1 + {x^2}}}{\rm{d}}x$ converges, by Theorem 9.4.4, the series also converges.
