TITLE: find Limit $a_n= \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+n}$
QUESTION [7 upvotes]: show sequence $a_n= \frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+n}$
converges to log2
my attempt:

sequence $a_n$ is monotonic increasing
0<$a_n$<1/2

how to find limit?

REPLY [5 votes]: $$
\begin{align}
\log(2)
&=\sum_{k=1}^\infty\frac{(-1)^{k-1}}k\\
&=\lim_{n\to\infty}\sum_{k=1}^{2n}\frac{(-1)^{n-1}}k\\
&=\lim_{n\to\infty}\left(\vphantom{\sum_{k=1}^n}\right.\overbrace{\sum_{k=1}^{2n}\frac1k}^{\substack{\text{sum of all}\\\text{terms}}}-2\overbrace{\sum_{k=1}^n\frac1{2k}}^{\substack{\text{sum of even}\\\text{terms}}}\left.\vphantom{\sum_{k=1}^n}\right)\\[6pt]
&=\lim_{n\to\infty}\left(\sum_{k=1}^{2n}\frac1k-\sum_{k=1}^n\frac1k\right)\\[6pt]
&=\lim_{n\to\infty}\sum_{k=n+1}^{2n}\frac1k\\
\end{align}
$$