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Suppose $s$ is not an integer, let $\lambda(s) = \min_{n\geq0}|s + n|$. Show that $\sum\limits_{n=1}^{\infty}(\frac{1}{n+s}-\frac{1}{n})\ll\frac{1}{\lambda(s)} + \log(|s| + 2)$.

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As stated, this looks to me like a homework problem. Is there any particular reason (if this is not homework) why you are considering this particular problem? Please check mathoverflow.net/faq#whatnot to to see if your question is appropriate for MO. If not, there are other sites listed at the link where you can ask this. – David Roberts Oct 3 2011 at 3:51
I confess that I could not get it straight when I read it from a book. – ksj03 Oct 3 2011 at 4:07
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 cross-posted math.stackexchange.com/questions/69442/… – Will Jagy Oct 3 2011 at 4:20
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 If it turns out that the poster has a problem understanding or beginning the demonstration, they should mention that and ask a more pointed question, rather than a close relative of "Solve this for me since I can't tell you yet what I don't understand." Even with such a clarification, I suspect the question is not suitable for MathOverflow. Gerhard "Ask Me About System Design" Paseman, 2011.10.02 – Gerhard Paseman Oct 3 2011 at 4:51
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 @ksj03: Just a hint: you will typically get a better reaction to your questions, both here and at math.stackexchange.com, if you phrase them as polite requests for assistance rather than using the imperative (show this, prove that, etc). – Neil Strickland Oct 3 2011 at 9:28
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closed as off topic by David Roberts, Will Jagy, algori, Dan Petersen, Igor Rivin Oct 3 2011 at 7:34

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