TITLE: Convergence of $\sum\frac{1}{n(\log(n))^c}$
QUESTION [5 upvotes]: Analyze the convergence of $$\sum\frac{1}{n(\log(n))^c}.$$
I know that it diverges for $c\leq1$ by using comparision test and integral test for convergence through the $$\int_{2}^{\infty}\frac{1}{x\log x}dx= \log(\log(\infty))-\log(\log2)$$
I also have a gut feeling that it converges for $c>1$ but not able to prove so. Please try explaining through less exotic methods by not resorting to weird convergence test theorem. Thanks!

REPLY [2 votes]: The integral test also works: by substitution, we have 
$$\int_{2}^\infty \frac{\mathrm d t}{t\,\ln^ct}=-\frac 1{(c-1)\ln^{c-1}t}\Biggr|_2^\infty=0+\frac1{(c-1)\ln^{c-1}2}\quad\text{ if }\;c>1.$$