Does that series converge or diverge?
Solution 1:
Calculate the sum, $n=\lceil e^k\rceil$ to $\lfloor e^{k+1}\rfloor$. There are about $(e-1)e^k$ terms. Each has the same sign, and each has absolute value $\ge \frac{1}{\sqrt{e^{k+1}}}$. So the Cauchy Criterion for the partial sums fails, and the series does not converge.