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Let $a_n$ be a sequence in (0,1) and consider $S_n=\sum_{k=1}^n a_k$ and $T_n=\sum_{k=1}^n S_k$. It can be seen that the series $$ \sum_{n=1}^\infty \frac{a_n}{S_n}\qquad\text{converges if and only if }\qquad \sum_{n=1}^\infty a_n\quad\text{ converges and} $$ $$ \sum_{n=1}^\infty \frac{a_n}{S_n^2}\qquad\text{always converges,} $$ What about the convergence of the series $$ \sum_{n=1}^\infty \frac{a_n}{T_n}\,? $$

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Since $S_n$ is increasing at at most linear rate, it is not too hard to show that $T_n \geq S_n^2/2,$ so this follows from your second fact (which seems hard to show without Stieltjes integration, unlike the first fact, which is quite easy directly).

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    $\begingroup$ Very good, thanks. The second fact follows by estimating with $\frac{a_n}{S_nS_{n-1}}=\frac{S_n-S_{n-1}}{S_nS_{n-1}}=\frac{1}{S_{n-1}}-\frac{1}{S_n}$ $\endgroup$ – fabrizio Dec 9 '13 at 9:38
  • $\begingroup$ Oh, duh (of course, gets the same answer as Stiltjes integration, but embarrassing nonetheless). $\endgroup$ – Igor Rivin Dec 9 '13 at 12:53

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