Is convergente or diverge?

Question

If $a_n>0, \ \forall n \in \mathbb{N}$ and the sum $\displaystyle\sum_{n=1}^{\infty}a_n$ diverges, then does $$\displaystyle\sum_{n=1}^{\infty}\frac{a_n}{(a_1+a_2+\cdots+a_n)^2},$$ converge?

Answer 1

Let $$S_n = \sum_{k=1}^{n}{a_k}$$ Now $$\sum_{n=1}^{\infty}\frac{a_n}{(a_1+a_2+\cdots+a_n)^2} = \frac{1}{a_1}+\sum_{n=2}^{\infty}{\frac{S_n - S_{n-1}}{S_n^2}}<\frac{1}{a_1}+\sum_{n=2}^{\infty}{\frac{S_n - S_{n-1}}{S_n S_{n-1}}}=\frac{1}{a_1}+\sum_{n=2}^{\infty}{\frac{1}{S_{n-1}}-\frac{1}{S_{n}}} = \frac{1}{a_1}+\frac{1}{S_1}=\frac{2}{a_1}$$ So therefore the sum is bounded above, so it must converge.