Let $\sum\_{n=1}^{\infty}{a\_n}$ be a series such that for each $n$, $a\_n>0$ and $(a\_n)$ is decreasing. Suppose that $\sum\_{n=1}^{\infty}{a\_{2n}}$ converges. Prove that $\sum\_{n=1}^{\infty}{a\_n}$ also converges. I try to prove by using definition but I got nowhere . Can anyone guide me ?