I'm looking for the limit of this sum: $\frac{1}{\left\lceil\frac{n}{2}\right\rceil+1}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+2}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+3}+\cdots+\frac{1}{n}$ when $n \rightarrow +\infty$. I guess the limit exists. But I don't know how to find it out. It seems that the formula $-\ln(1-x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\cdots+\frac{x^n}{n}+\cdots$ doesn't help. May I have some help?
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This is related to this question, and may be a duplicate of this question. – robjohn Aug 29 '15 at 06:58
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Your partial sum is $H_n-H_{\left\lceil\frac{n}{2}\right\rceil}$, where $H_n$ is the $n^{\rm{th}}$ harmonic number Since $H_n \approx \log n + \gamma,$ this will approach $\log n - \log {\left\lceil\frac{n}{2}\right\rceil}=\log 2$
Ross Millikan
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@Connor: yes, I dropped a sign. A $\LaTeX$ hint: if you put a backslash before ln, you get the right font, so \ln instead of ln gives $\ln$ instead of $ln$ – Ross Millikan Aug 29 '15 at 01:37