Answer
Diverges.
Work Step by Step
$\lim\limits_{n \to \infty}a_{n} = \lim\limits_{n \to \infty} \frac{n^{2}cosn}{1+n^{2}}$
$=\lim\limits_{n \to \infty} \frac{\frac{n^{2}cosn}{n^{2}}}{\frac{1+n^{2}}{n^{2}}}$
$=\frac{\lim\limits_{n \to \infty}cosn}{\lim\limits_{n \to \infty} \frac{1}{n^{2}}+1}$
$=\frac{\lim\limits_{n \to \infty}cosn}{1}$
$\lim\limits_{n \to \infty}cosn$ does not exist because $cosn$ is a periodic function.
Therefore, the given sequence diverges.
