Answer
Convergent
Work Step by Step
$\Sigma_{n=1}^{\infty} \frac{\sqrt n+4}{n^{2}}=\Sigma_{n=1}^{\infty} \frac{\sqrt n}{n^{2}}+\Sigma_{n=1}^{\infty} \frac{4}{n^{2}}$
Here,
$\Sigma_{n=1}^{\infty} \frac{\sqrt n}{n^{2}} =\Sigma_{n=1}^{\infty} \frac{1}{n^{3/2}}= $convergent by p-series
and
$\Sigma_{n=1}^{\infty} \frac{4}{n^{2}}$= convergent by p-series.
Hence, the given series is convergent.
