TITLE: A series involving $\prod_1^n k^k$
QUESTION [8 upvotes]: Is this series $$\sum_{n\geq 1}\left(\prod_{k=1}^{n}k^k\right)^{\!-\frac{4}{n^2}} $$ convergent or divergent?
My attempt was to use the comparison test, but I'm stuck at finding the behaviour of $\displaystyle \prod_1^n k^k$ as $n$ goes to infinity. Thanks in advance.

REPLY [4 votes]: By Cesaro-Stolz theorem, we have that $$\lim_{n\to\infty}\frac{1}{n^2 \log(n)}\sum_{k=1}^{n} k\log(k)=\frac{1}{2}$$ and then we see   that for $n$ large, we have that $$\frac{1}{n^2}\sum_{k=1}^{n} k\log(k)\approx\frac{\log(n)}{2}\Rightarrow \frac{-4}{n^2}\sum_{k=1}^{n} k\log(k)\approx -2\log(n)$$and thus $$\left(\prod_{k=1}^{n}k^k\right)^{\!-\frac{4}{n^2}}\approx \frac{1}{n^2}$$
whence we conclude the series converges.
Q.E.D. (an elementary way , without integrals)