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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 an important example because it tells us that there exists no series which diverge slowest. So I want to verify this fact in different aspect.

I know a method which uses Cauchy's Theorem. For any $n \in \bf N$ , choose a sufficiently 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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    $\begingroup$ That's probably the simplist way. See this also (I assume this is the argument you're using). $\endgroup$ – David Mitra Jul 15 '13 at 12:27
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    $\begingroup$ 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 $\sqrt{S_n}$. $\endgroup$ – Rodrigo Apr 30 '14 at 4:57
  • $\begingroup$ Is that answer your question? $\endgroup$ – user146010 Sep 11 '14 at 3:25
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First : $a_n= S_n - S_{n-1}$

Let $U_n$ be : $U_n = \frac{S_n- S_{n-1}}{S_n} = 1 - \frac{S_{n-1}}{S_n}$

Then : $\ln(S_n) -\ln(S_{n-1}) = -\ln(1-U_n)$

Suppose the series of general term $(U_n)$ converges:

$\implies (U_n) \rightarrow 0$

$\implies$ $\ln(S_n) - \ln(S_{n-1}) = U_n + o(U_n) $

It means that $( \ln(S_n) - \ln(S_{n-1}) )$ is the term of a convergent series, thus implying that $(S_n)$ is a converging sequence, which is absurd.

Hence the series of general term $(U_n)$ diverges, always.

Now one interesting question is: what c does make $U_n(c) = \frac{a_n}{S_n^c}$ converge? You know that $c \geq 1$ , but is it the inf?

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    $\begingroup$ Downvotes and no comments, classic math.exchange. Deontology still not a thing here, shameful really . The guy who pointed to the proof of op's very result got 4 upvotes for his comment, I give another solution and a generalization of his example and I get 2 downvotes. You guys actually achieve to turn maths into mediocrity, I have to say I'm baffled but not positively $\endgroup$ – mvggz Aug 11 '16 at 9:14
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    $\begingroup$ Insulting everyone for an opinion of one person is neither productive, nor helpful, nor it stimulates upvotes. If you suspect that foul play is involved, you can always flag your answer for moderator attention. $\endgroup$ – TZakrevskiy Aug 13 '16 at 18:14
  • $\begingroup$ Note that upvotes on comments are very different from upvotes on answers. Also, I think the reason that comment was upvoted was to indicate agreement with the claim "that's probably the simplest way". $\endgroup$ – Noah Schweber Aug 13 '16 at 18:40
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    $\begingroup$ And what's the point of downvoting an answer if there is not one single information about why. Just what's the point.. And you don't even seem to find it the most troubling thing altogether, that comforts me about the seriousness of this community. If this site is not about pedagogy and emulation, then it's useless just like downvoting an answer with not a single explanation. I did not believe it was possible to disgust me of mathematical material, but this community actually achieved that. That's not how I 'exchange' about math with someone, I leave that to all of you $\endgroup$ – mvggz Aug 15 '16 at 15:16

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