# series convergence

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.

Yaron.

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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
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 $n>N$...
@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