Convergence of a series $\sum a_n/T_n$ where $T_n=\sum S_k$ and $S_n=\sum a_k$

Question

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 $$ \sum_{n=1}^\infty \frac{a_n}{T_n}\,? $$

Answer 1

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).