# Convergence of $\sum_{k=0}^n 1+\left(k+\frac{1}{2}\right)\ln\left(\frac{k}{k+1}\right)$

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.

• 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. – Jack D'Aurizio Apr 28 at 21:20

$$a_n=1-\left(n+\frac12\right)\log\left(1+\frac1n\right)=1-\left(n+\frac12\right)\left(\frac1n-\frac1{2n^2}+\frac1{3n^3}\right)+o(\frac{1}{n^2})$$ I let you expand the expression above.