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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?

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marked as duplicate by Old John, Nameless, Marvis, amWhy, Mike Spivey Dec 23 '12 at 21:15

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@experimentX it has to have the word converges two times –  Jorge Fernández Dec 23 '12 at 20:56

1 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.

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