Using definite integral to find the limit.

Question

I would like someone to verify this exercise for me. Please.

Find the following limit:

$\lim\limits_{n \to \infty}\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n}\right)$

$=\lim\limits_{n \to \infty}\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{n+2n}\right)$

$=\lim\limits_{n \to \infty}\sum\limits_{k=1}^{2n} \dfrac{1}{n+k}$

$=\lim\limits_{n \to \infty}\sum\limits_{k=1}^{2n} \dfrac{1}{n\left(1+\frac{k}{n}\right)}$

$=\lim\limits_{n \to \infty}\sum\limits_{k=1}^{2n}\left(\dfrac{1}{1+\frac{k}{n}}\cdot\dfrac{1}{n}\right)$

$=\displaystyle\int_{1+0}^{1+2} \frac{1}{x} \,dx$

$=\displaystyle\int_{1}^{3} \frac{1}{x} \,dx$

$=\big[\ln|x|\big] _{1}^3$

$=\ln|3|-\ln|1|$

$=\ln(3)-\ln(1)$

$=\ln(3)$

Answer 1

Lastly, use $\;k/n\rightarrow x,\;\;1/n \to dx\;,\;$ then $$L=\int_{0}^{2} \frac{dx}{1+x}=\ln(1+x)\big|_{0}^{2}=\ln 3\;.$$

Edit: the lower limit is $x_l=1/n, x_u=2n/n$, when $n$ is large ($\infty$} these are 0 and 2

Answer 2

\begin{align}S_n&=\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n}\\ &=\sum_{k=1}^{2n}\frac{1}{n+k}\\ &=\sum_{k=1}^{n}\frac{1}{n+k}+\underbrace{\sum_{k=n+1}^{2n}\frac{1}{n+k}}_{l=k-n}\\ &=\sum_{k=1}^{n}\frac{1}{n+k}+\sum_{k=1}^{n}\frac{1}{2n+l}\\ &=\sum_{k=1}^{n}\left(\frac{1}{n+k}+\frac{1}{2n+k}\right)\\ &=\frac{1}{n}\sum_{k=1}^{n}\left(\frac{1}{1+\frac{k}{n}}+\frac{1}{2+\frac{k}{n}}\right)\\ \end{align} Therefore, \begin{align}\lim_{n\rightarrow \infty}S_n&=\int_0^1 \left(\frac{1}{1+x}+\frac{1}{2+x}\right)dx\\ &=\Big[\ln(1+x)+\ln(2+x)\Big]_0^1\\ &=\ln 2+\ln 3-0-\ln 2\\ &=\boxed{\ln 3} \end{align}