Answer
Convergent
Work Step by Step
$\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\dfrac{\frac{(n+1)^{2}+1}{5^{n+1}}}{\frac{n^{2}+1}{5^{n}}}|$
$=\frac{1}{5}\lim\limits_{n \to \infty}|\frac{n^{2}+2n+2}{n^{2}+1}|$
$=\frac{1}{5}\lim\limits_{n \to \infty}|\frac{1+2/n+2/n^{2}}{1+1/n^{2}}|$
$=\frac{1}{5}\times \frac{1+0+0}{1+0}$
$=\frac{1}{5} \lt 1$
The series is convergent.