CURED - 2015-11-29

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TheGreatDuck

2:05 AM

But the question is vague and nonprecise

It looks like one I'd delete

Normal Human

Well, it should be closed then. If there is an answer, there is no way to get rid of the question besides closing it.

5 hours later…

SmokeDetector

7:03 AM

[ SmokeDetector ] Few unique characters in answer: Define function J by kristin on math.stackexchange.com

2 hours later…

Jesse P Francis

9:21 AM

I dodn't notice the answer there. Actually it is a duplicate; I flagged it and it hasn't see light for past one week.

Q: If $a_n$ is a strictly increasing unbounded sequence, does $\sum_n \frac{a_{n+1} - a_n}{a_n}$ diverge?

So I've been thinking through some test cases. If $a_n = n$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ is the harmonic series which diverges. And if $a_n = \sum_{k=1}^n 1/k$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ diverges like $\sum_n 1/(n \log n)$. So that got me thinking, if $a_n$ is a strictly i...