Convergent $\sum_{n=2}^\infty \frac 1 {n \log^n(n)}$
I have to prove that this series is convergent:
$$\sum_{n=2}^\infty \frac 1 {n \log^n(n)}$$
I'm having problems with the limit for the ratio test, and I've already tried the comparision test and to integrate the function.
Any hints?
Thanks
Try: $$\frac{1}{n\log^n n}< \frac{1}{2^n}$$ for $n>e^2$.
Hint: comparison test with $$ \sum_{n=2}^\infty \frac{1}{n\log^2(n)} $$ In order to show that $\sum_{n=2}^\infty \frac{1}{n\log^2(n)}$ converges, use the integral test.
Hint for the integral: $u$ substitution with $u = \log(x)$