Prove that if $\sum |a_n|$ converges, then $\sum a_n^2$ also converges.


$\sum |a_n|$ converges $\implies \sum |a_n|<M$.

If $\sum |a_n|$ converges, then $\sum a_n $ also converges.

Could anyone please tell me on how to proceed ahead with regard to this problem.

Thank you very much for your help


marked as duplicate by Git Gud, copper.hat, leo, Lord_Farin, user98602 Jan 26 '15 at 18:18

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  • 1
    $\begingroup$ $\sum a_n^2 \le (\sum |a_n|)^2$,bounded monotone convergence thm $\endgroup$ – r9m Jan 26 '15 at 17:48
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    $\begingroup$ After a while $|a_n|\lt 1$, so $a_n^2\lt |a_n|$. $\endgroup$ – André Nicolas Jan 26 '15 at 17:49
  • $\begingroup$ Thank you for your answers. I just lost it all in this problem, I guess. $\endgroup$ – MathMan Jan 26 '15 at 17:52
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    $\begingroup$ Note that $|a_n|$ must be bounded by some $M$ and so $|a_n|^2 \le M |a_n|$. $\endgroup$ – copper.hat Jan 26 '15 at 17:52

Prove that there is a tail of the summation for which $a_n^2<|a_n|$ holds for all terms in the tail.


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