Answer
Convergent
Work Step by Step
In the given problem
$a_{n}=\frac{n^{2n}}{(1+2n^{2})^{2}}$
$\frac{n^{2n}}{(1+2n^{2})^{2}}=(\frac{n^{2}}{(1+2n^{2}})^{2}$
Therefore,
$\lim\limits_{n \to \infty}|a_{n}|^{1/n}=\lim\limits_{n \to \infty}\frac{n^{2}}{1+2n^{2}}$
Divide numerator and denominator by $n^{2}$
$=\lim\limits_{n \to \infty}\frac{1}{\frac{1}{n^{2}}+2}$
$=\frac{1}{0+2}$
$=\frac{1}{2} \lt 1 $
Hence, the series converges by the Root Test.
