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i ran into this question:

prove or show false:

if $\sum_{n=1}^{\infty}a_{n}$ is a converging series, but the series $\sum_{n=1}^{\infty}a_{n}^2$ diverges, then $\sum_{n=1}^{\infty}a_{n}$ is conditionally convergent.

I'm pretty sure it's true because I couldn't find any example that shows otherwise, but still, i can't find the proof.

Thanks in advance,


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Hint: If $\sum|a_n|$ converges, then $\sum a_n^2$ converges (by the Comparision Test). – David Mitra May 19 '13 at 10:59
up vote 2 down vote accepted

We want to show that: if the series $\sum_{n=1}^{\infty}a_{n}^2$ diverges then the $\sum_{n=1}^{\infty}|a_{n}|$ diverges.

To prove this use that $a^2_n<1$ for all $n>N$ for some $N\in\mathbb N$ and therefore $|a_n|>a_n^2$ for all $n>N$...

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no, thi is not what we want to show. – user76508 May 19 '13 at 11:02
@user76508 Yes it is. (Note that P. implicitly made use of the convergence of $\sum a_n$ to show that $a_n^2<1$ for almost all $n$) – Hagen von Eitzen May 19 '13 at 11:08
@user76508 It actually is what you're trying to show; since divergence of the sum absolute values is definition of conditional convergence (of course, assuming the original series converges). – Peter Košinár May 19 '13 at 11:08
got it, thank you – user76508 May 19 '13 at 11:10

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