TITLE: Prove or disprove convergence of a series
QUESTION [5 upvotes]: Assume  $\sum_{n=k}^{\infty} a_n$ converges.
Then $\sum_{n=k}^{\infty} a_n \cdot a_{n+2}$ converges?
I think I should disprove it, because $\sum_{n=k}^{\infty} a_{n+2}$ might diverge but I can't find a disproof.

REPLY [22 votes]: Hint. Consider the sequence $a_n=\frac{(-1)^n}{\sqrt{n}}$ for $n\geq 1$.