# Prove $\sum _{n=1}^{\infty }\left(\frac{1}{n}-\ln\left(\frac{\left(1+n\right)}{n}\right)\right)\:$ converges

Prove $\sum _{n=1}^{\infty }\left(\frac{1}{n}-\ln\left(\frac{\left(1+n\right)}{n}\right)\right)\:$ converges

I tried using taylor expansion and Cauchy condensation test, but got stuck.

Any help appreciated.

• That is (one of) the definition(s) of Euler-Mascheroni constant. – Jack D'Aurizio Mar 6 '17 at 15:49

By the Taylor expansion of the logarithm, for $n\geq 1$: $$\ln\left(\frac{n+1}{n}\right) = \frac{1}{n} +O(n^{-2}).$$ The result follows.
Hint: $\lim\limits_{x\to0}\frac{x-\ln(1+x)}{x^2}=\frac12$