If $\sum \limits_{n=1}^\infty a_n$ diverges, then does $\sum \limits_{n=1}^\infty a_{2n}$ diverge too?
$a_n>0$ and $(a_n)$ is decreasing. Suppose that $\sum_{n=1}^{\infty}{a_{2n}}$ converges. Prove that $\sum_{n=1}^{\infty}{a_n}$ also converges.