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Let $a_n>0$ and $S_n=\sum_{k=1}^{n}a_n$. If $\lim_{n \rightarrow \infty}S_n = +\infty$, then $\sum_{n=1}^{\infty}\frac{a_n}{S_n}=+\infty$. I think this is a important example because it tell us that there exists no series which diverge slowest. So I want to verify this fact in different aspect.

I know a method which use Cauchy's Theorem. for any $n \in \bf N$ . choose a sufficient large $p \in \bf{N}$. we have $$\sum_{k=n+1}^{n+p}\frac{a_k}{S_k}\geq \frac{S_{n+p}-S_{n}}{S_{n+p}}\geq \frac{1}{2}$$


Is there any other approach to it? thanks very much.

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That's probably the simplist way. See this also (I assume this is the argument you're using). –  David Mitra Jul 15 '13 at 12:27
1  
If you just want to prove that there is no slowest diverging series, you can see that you can actually choose the increasing sequence $S_n$, and define $a_n=S_n-S_{n-1}$. Then, for any sequence $S_n$ you can get the slower diverging sequence $\sqrt{S_n}$. –  Rodrigo Apr 30 at 4:57

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