TITLE: Proving the convergence of a series
QUESTION [8 upvotes]: I'm trying to solve this exercise:

Let $(a_n)_{n\in\mathbb{N}}\subseteq\mathbb{R}$ and $p,q\in\mathbb{R}$ such that $p<q$. Prove that if the series $\sum \frac{a_n}{n^p}$ converges, then the series $\sum \frac{a_n}{n^q}$ also does.

The comparison test sounds useful, but since we don't know that $a_n$ is positive it doesn't apply. Any pointers?

REPLY [3 votes]: If $\displaystyle\sum_{n=1}^\infty\frac{a_n}{n^p}$ converges, we know that
$\displaystyle\sum_{n=1}^m\frac{a_n}{n^p}$ is bounded. Furthermore, $\dfrac{1}{n^{q-p}}$ decreases monotonically to $0$. Thus, Dirichlet's Test says that
$$
\sum_{n=1}^\infty\frac{a_n}{n^q}=\sum_{n=1}^\infty\frac{a_n}{n^p}\frac{1}{n^{q-p}}
$$
converges.