_n[11+n+12+n+13+n++12n] is equal to:

Question

$\displaystyle\lim_{n	o\infty}\left[\frac1{1+n}+\frac1{2+n}+\frac1{3+n}+\cdots+\frac1{2n}ight]$ is equal to:

Accepted Answer

$$
\begin{aligned}
& \lim _{n \rightarrow \infty}\left[\frac{1}{1+n}+\frac{1}{2+n}+\frac{1}{3+n}+\ldots \ldots+\frac{1}{2 n}\right] \\
& =\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{r+n} \\
& =\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{n}\left(\frac{1}{\frac{r}{n}+1}\right) \\
& =\int_0^1 \frac{d x}{x+1} \\
& =\left.\log _e(1+\mathrm{x})\right|_0 ^1 \\
& =\log _e^2
\end{aligned}
$$

Solutions · 1