Len $a_n$ be infinite sequence such that $|{a_{n+1}}|<|a_n|$. Assumbe that $S_{2^n}$ is bounded. Does it imply that $S_n$ is bounded too?

Asked Oct 20 '14 at 09:57

Active Oct 20 '14 at 11:06

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Len $a_n$ be infinite sequence such that $|{a_{n+1}}|<|a_n|$ and let $S_n = \sum\limits_{i=1}^n{a_i}$. Assumbe that $S_{2^n}$ is bounded, i.e. there exists positive $B$ such that for any natural $n$ we have $|S_{2^n}|<B$. Does it imply that $S_n$ is bounded too?

asked Oct 20 '14 at 09:57

Hedgehog

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See Cauchy condensation test. – Eclipse Sun Oct 20 '14 at 10:03
@Eclipse Sun This series is not positive and I do not ask about convergence. – Hedgehog Oct 20 '14 at 10:14
The method is nearly the same. – Eclipse Sun Oct 20 '14 at 14:43

Len $a_n$ be infinite sequence such that $|{a_{n+1}}|<|a_n|$. Assumbe that $S_{2^n}$ is bounded. Does it imply that $S_n$ is bounded too?

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