TITLE: For $a_n>0$ such that $\sum a_n $ converges, show that there exist $c_n>0$ such that $c_n\to \infty$ and $\sum a_n c_n$ is finite.
QUESTION [10 upvotes]: A problem From PhD Pre lims Exam:

Let $ a_{n} > 0 $ for all $ n\in\mathbb{N}$ such that $\sum a_{n} $ converges. Show that there exist $ c_{n} > 0 $ ($n\in\mathbb{N}$) such that $ \lim \limits_{n\to\infty} c_{n}= \infty $ and $\sum a_{n}c_{n} $ is finite.

REPLY [23 votes]: Let 
$$r_n=\sum_{k=n+1}^\infty a_n$$
so since the series $\sum a_n$ is convergent then the sequence $(r_n)$ is decreasing and convergent to $0$.
We have
$$\frac{a_n}{\sqrt{r_{n-1}}}=\frac{r_{n-1}-r_n}{\sqrt{r_{n-1}}}=\frac{(\sqrt{r_{n-1}}-\sqrt{r_{n}})(\sqrt{r_{n-1}}+\sqrt{r_{n}})}{\sqrt{r_{n-1}}}\leq2(\sqrt{r_{n-1}}-\sqrt{r_{n}})=t_n$$
and since the series $\sum t_n$ is convergent (telescoping series) then the series 
$$\sum \frac{a_n}{\sqrt{r_{n-1}}}$$
is convergent. Take $c_n=\frac{1}{\sqrt{r_{n-1}}}$.