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

Question

Attempt:

$\sum |a_n|$ 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

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

score 4 · Answer 1 · answered Jan 26 '15 at 17:49

4

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

answered Jan 26 '15 at 17:49

T.J. Gaffney

1 Answers1