Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
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
add comment

Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.