I found this question on another forum so I don't have more details about it other than it seems non-intuitive to me:
Let $\sum_{n=1}^\infty a_n$ consist of positive terms and be convergent. Define $b_n = \frac{1}{na_n^2}$. Is the series $\sum_{n=1}^\infty \frac{n}{\sum_{j=1}^n b_j}$ also convergent?
The place where I found the question claims the statement is true, but I have my doubts. Any ideas?