_ n → ∞ ((1)/(1 + n) + (1)/(2 + n) + (1)/(3 + n) + + (1)/(2n)) is...

We have,

$\lim_{n \to \infty}\left(\dfrac{1}{1 + n} + \dfrac{1}{2 + n} + \dfrac{1}{3 + n} + \ldots + \dfrac{1}{2n}\right)$

$= \lim_{n \to \infty}\left(\dfrac{1}{1 + n} + \dfrac{2}{1 + n} + \dfrac{3}{1 + n} + \ldots . + \dfrac{1}{n + n}\right)$

$= \lim_{n \to \infty}\mathop{∑}_{r = 1}^{n}\dfrac{1}{n + r}$

$= \lim_{n \to \infty}\mathop{∑}_{r = 1}^{n}\dfrac{1}{n}\left(\dfrac{1}{1 + \dfrac{r}{n}}\right)$

$= {∫}_{0}^{1}\dfrac{1}{1 + x}dx$

$= {\left[\log_{e}\left(1 + x\right)\right]}_{0}^{1}$

$= \log_{e}\left(2\right)$