Let $0 < a < 1$. I'm trying to figure out whether the following series converges:
$\sum_{k=1}^\infty k^a \frac{1}{k(k+1)}.$
Now it's clear that if $a$ were greater than or equal to $1$ then this series would diverge since
$\sum_{k=1}^\infty k^1 \frac{1}{k(k+1)} = \sum_{k=1}^\infty \frac{1}{(k+1)} = \sum_{k=2}^\infty \frac{1}{k} = \infty.$
So this makes it a bit hard to think of a bound for the series in question. Any advice?