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The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$

What is the sum of the 'second half' of the harmonic series? $$\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} =~ ?$$ More precisely, what is the limit of the above sequence of partial ...
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Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

Inadvertently, I find this interesting inequality. But this problem have nice solution? prove that $$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$ This problem have nice solution? Thank you. ago,I find ...
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It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$ If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$ Starting: $$\... 7answers 28k views Summing Finitely Many Terms of Harmonic Series: \sum_{k=a}^{b} \frac{1}{k} How do I calculate sum of a finite harmonic series of the form :$$\sum_{k=a}^{b} \frac{1}{k} = \frac{1}{a} + \frac{1}{a+1} + \frac{1}{a+2} +\cdots \frac{1}{b}.$$Is there a general formula for this?... 4answers 721 views Series for logarithms This is more of a challenge than a question, but I thought I'd share anyway. Prove the following identities, and prove that the pattern continues. \begin{equation*} \sum_{n=0}^\infty\left(\frac{1}{2n+... 5answers 692 views Can we assign a value to the sum of the reciprocals of the natural numbers? I know the sum of the reciprocals of the natural numbers diverges to infinity, but I want to know what value can be assigned to it.$$\sum_{n=1}^{\infty}\frac1n=\frac11+\frac12+\frac13+\frac14+\dots=...
This thread reminded me of an old unsettled question I have. Given an arbitrary conditionally convergent series $\beta=\sum\limits_{k=1}^\infty a_k$ and a target value $\alpha$, is there an algorithm ...