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Prove $$\sum{a_n}$$ converges if and only if $$\sum 2^n a_{2^n}$$ converges.

This is assuming that $a_n$ is decreasing.

user106279
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  • Hi, and welcome! I've edited your post to hopefully fix the subscript error you mentioned; please check that it's correct. Can you also share your thoughts on the problem? As it's stated, it's false - just choose any non-convergent sequence that has zeros for the terms corresponding to index $2^n$. –  Nov 07 '13 at 23:35
  • Are you assuming that the terms are decreasing? – bof Nov 07 '13 at 23:36
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    What you're looking for is almost certainly the Cauchy Condensation Test (http://en.wikipedia.org/wiki/Cauchy_condensation_test ) – Steven Stadnicki Nov 07 '13 at 23:41

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Consider the sequence

$$a_n = \left\{ \begin{array}{lr} 0 :& n = 2^k \text{ for some } k \\ 1 :& \text{ else } \end{array}\right.$$

As Steven Stadnicki commented, you're probably after the Cauchy condensation test, a proof of which is in the linked article. Note that $a_n$ fails this test because it's not non-increasing.