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I need to prove that this series converges as n goes to infinity: $$\sum_{k=1}^n \left( 1+\left(k+\frac{1}{2}\right) \ln\left(\frac{k}{k+1}\right) \right)$$ I tried to do a Taylor expansion of the logarithm but I end up with 2 sequences of opposite signs and I can't conclude on the convergence. Any help would be greatly appreciated.

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  • $\begingroup$ Probably the sum should start at $k=1$. $$\int_{1}^{+\infty}\left[\{x\}-\frac{1}{2}\right]\frac{dx}{x}$$ converges by integration by parts. $\endgroup$ – Jack D'Aurizio Apr 28 at 21:20
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$$a_n=1-\left(n+\frac12\right)\log\left(1+\frac1n\right)=1-\left(n+\frac12\right)\left(\frac1n-\frac1{2n^2}+\frac1{3n^3}\right)+o(\frac{1}{n^2})$$ I let you expand the expression above.

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  • $\begingroup$ oh god thank you, flipping the fraction in the logarithm is clever :) $\endgroup$ – Cominou Apr 28 at 17:57

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