Answer
The series $\sum_{n=1}^{\infty}\frac{2+(-1)^n}{n^{3/2} } $ converges.
Work Step by Step
We have the series:
$\sum_{n=1}^{\infty}\frac{2+(-1)^n}{n^{3/2} } $
Note that when $n$ is odd, then the series becomes $\frac{2-1}{n^{3/2}}=\frac{1}{n^{3/2}}$, a convergent p-series with $p=3/2\gt 1$. If $n$ is even, then we get $\frac{2+1}{n^{3/2}}=\frac{3}{n^{3/2}}$, which is also convergent. Thus, our series converges as well.
