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.

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    $\begingroup$ 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


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.


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