6
$\begingroup$

Assume that $a_n \geq 0$ , and that $a_n \to 0$ and $s_n = \sum_{k=1}^{n} a_k$ and that $\sum_{n=1}^{\infty} a_n = \infty$, prove that $\sum_{n=1}^{\infty} \frac{a_n}{s_n^2} $ is convergent!

This is not a homework, but the lecturer wrote it down as sort of a question that might show up in the test in a few weeks.

$\endgroup$
  • 1
    $\begingroup$ you need $a_1\ne 0$ $\endgroup$ – Exodd Dec 16 '17 at 19:40
  • $\begingroup$ You have a $\frac 00$ indetermined form in your series $\endgroup$ – Exodd Dec 16 '17 at 19:45
  • $\begingroup$ @Exodd you are right, actually we need that $a_1 >0$, thanks. $\endgroup$ – Ahmad Dec 16 '17 at 19:46
13
$\begingroup$

Hint

Use this inequality

$$\frac{a_n}{s_n^2}\le \frac{a_n}{s_ns_{n-1}}=\frac{s_n-s_{n-1}}{s_ns_{n-1}}=\frac{1}{s_{n-1}}-\frac{1}{s_{n}}$$ together with telescoping and Cauchy's criteria.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.