Answer
Divergent.
Work Step by Step
Notice that the series is not decreasing as is shown in the figure attached. Therefore, we cannot apply the Alternating Series Test. However, we can apply the Ratio Test for Absolute Convergence by evaluating:
$\rho = \mathop {\lim }\limits_{k \to \infty } \dfrac{{\left| {{a_{k + 1}}} \right|}}{{\left| {{a_k}} \right|}} = \mathop {\lim }\limits_{k \to \infty } \dfrac{{{3^{2k + 1}}}}{{{{\left( {k + 1} \right)}^2} + 1}}\cdot\dfrac{{{k^2} + 1}}{{{3^{2k - 1}}}} = 9\mathop {\lim }\limits_{k \to \infty } \dfrac{{{k^2} + 1}}{{{{\left( {k + 1} \right)}^2} + 1}}$
$ = 9\mathop {\lim }\limits_{k \to \infty } \dfrac{{1 + \dfrac{1}{{{k^2}}}}}{{{{\left( {1 + \dfrac{1}{k}} \right)}^2} + \dfrac{1}{{{k^2}}}}} = 9$
Since $\rho = 9 \gt 1$, by the Ratio Test for Absolute Convergence the series diverges.
