The problem is the following:
If $\sum \limits_{n=1} ^{\infty} a_n$ converges, where $a_n$ are real numbers, then there exists $b_n \to \infty$ so that $\sum \limits_{n=1} ^{\infty} a_n b_n$ is still convergent.
I know that the above statement is true if $a_n$ are nonnegative (setting $b_n=\frac1{\sqrt{R_{n-1}}+ \sqrt{R_{n}}}$, where $R_n=\sum \limits_{k=n} ^{\infty} a_k$). But for the general $a_n$, I have no idea how to prove it.
Any ideas are welcome. Thanks!