I have already proved that the series above does not converge absolutely. However, I am not sure how to check whether the series converges conditionally.
I thought about using the Alternating Series Test, but I'm not sure if I can because I don't think that the sequence $\frac{1}{n^{1+\frac{1}{n}}}$ is monotonically decreasing.
If that is the case, how should I approach showing whether the series converges conditionally or diverges? Any guidance would be appreciated.
$n^{1+1/n} = \exp \left( \left(1+\frac{1}{n} \right)\ln n \right)$ and you can see that $ \left(1+\frac{1}{n} \right)\ln n$ increases when $n\ge 2$, therefore the sequence $1/n^{1+1/n}$ decreases eventually and so you can apply alternating series test. When applying series test, first few terms are not affect the convergence of the series so you can ignore finitely many terms.