Answer
Divergent
Work Step by Step
Given: $\Sigma_{n =1}^{ \infty}\frac{1}{n+ncos^{2}n}$
$cosn \leq 1$
$ncos^{2} n \leq n$
$n+ncos^{2}n\leq n+n$
$\frac{1}{n+ncos^{2}n}\geq \frac{1}{2n}$
$\Sigma_{n =1}^{ \infty}\frac{1}{n+ncos^{2}n} \geq \frac{1}{2}\Sigma_{n =1}^{ \infty} \frac{1}{n}$
The given series is greater than $\frac{1}{2}$ times the harmonic series. The harmonic series is divergent.
Hence, the given series is also divergent.