The last problem I posted had a wrong statement. I recovered the correct one.
Let $(a_n)$ be a sequence of positive real numbers.
Prove that $\sum_{n\geq 0}a_n$ converges iff $\displaystyle \sum_{n\geq 0} \frac{a_n}{\sum_{k=0}^n a_k}$ converges
The direct statement is easy to prove using comparison test.
I'm (again!) stuck with the converse.
I tried summation by part, without success.
Note that the convergence of $\displaystyle \sum_{n\geq 0} \frac{a_n}{\sum_{k=0}^n a_k}$ implies $\displaystyle \frac{a_n}{\sum_{k=0}^n a_k} \to 0$
I don't know what to do next ...