Suppose that $a_k$ are positive and decreasing. Prove that $\sum_{k=1}^{\infty}(a_k)$ if and only if $\sum_{k=1}^{\infty}{2^ka_{2^k}}$ converges. [duplicate]

Question

Possible Duplicate:
proving cauchy condensation test

Suppose that $a_k$ are positive and decreasing. Prove that $\sum_{k=1}^{\infty}(a_k)$ if and only if $\sum_{k=1}^{\infty}{2^ka_{2^k}}$ converges.

By using decreasing how can I prove this?

Answer 1

This is the Cauchy Condensation test for convergence. Wikipedia has a decent page on it, and it is also covered in many textbooks that cover convergence of series.