TITLE: Does $\sum^{\infty}_{n=1}\frac{\ln{n}}{n^{1.1}}$ converge or diverge?
QUESTION [5 upvotes]: Does $$\sum^{\infty}_{n=1}\frac{\ln{n}}{n^{1.1}}$$ converge or diverge? I think the basic comparison works but I have a hard time finding a comparer. Could someone suggest one?

REPLY [2 votes]: One has, by the integral convergence test and an integration by parts:
$$
0<\sum^{\infty}_{n=1}\frac{\ln{n}}{n^{1.1}}\leq \int^{\infty}_{1}\frac{\ln{x}}{x^{1.1}}dx=\left[-\frac{\ln x}{x^{0.1}}\right]^{\infty}_{1}+\int^{\infty}_{1}\frac{dx}{x^{1.1}}=100
$$ the series is convergent.