I need help proving the following:

If $\sum_{n=1}^\infty a(n)$ converges:

prove $$\sum_{n=1}^\infty \frac{a(n)}{n^2}$$ converges absolutely.

I managed to prove that the series converge using Abel's test but I can't find a way to show it's converging absolutely.

  • 4
    $\begingroup$ If $\sum a_n$ converges, the sequence $(a_n)$ is bounded. $\endgroup$ Jun 1 '17 at 5:29
  • $\begingroup$ @LordSharktheUnknown how that helps proving? $\endgroup$
    – segevp
    Jun 1 '17 at 5:31

As per Lord Shark comment,If $\sum{a_{n}}$ converges then the sequence $a_{n}$ is a bounded sequence which means that there exists $M$ such that $|a_{n}| \leq M \forall n \in \mathbb{N}$,so for dealing with the absolute convergence we take the absolute value of the sequential terms as

$\sum{|\frac{a_{n}}{n^2}|} \leq \sum {\frac{M}{n^2}}$ , now we know that the right side sum converges and by comparision test we can say that the absolute convergence is obtained!

  • $\begingroup$ You are welcome! all the best :) $\endgroup$
    – BAYMAX
    Jun 1 '17 at 6:56

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