Question

Check whether the sequence (an), where

an = 1/ (n+1) + 1/(n+2) +....+1/(2n) is convergent or not

Answer 1

It converges to $\ln2$.

The sequence $x_n$ is $H_{2n}-H_n$, where $H_n$ is the $n^{th}$ harmonic number. It is known that $\lim\limits_{n\to+\infty}(H_n-\ln n)$ is $\gamma$, the Euler-Mascheroni constant. Hence

$\lim\limits_{n\to\infty}((H_{2n}-\ln2n)-(H_n-\ln n))=\gamma-\gamma=0$.

$\lim\limits(H_{2n}-H_n)=\ln2n-\ln n=\ln\frac{2n}{n}=\ln2$.