1. Tardigrade
  2. Question
  3. Mathematics
  4. displaystyle limn → ∞[(1/2n+1)+(1/2n+2)+.....+(1/2n+n)]=

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. $\displaystyle \lim_{n \to \infty}$$\left[\frac{1}{2n+1}+\frac{1}{2n+2}+.....+\frac{1}{2n+n}\right]=$

Integrals

Solution:

$\displaystyle \lim_{n \to \infty}$$\left[\frac{1}{2n+1}+\frac{1}{2n+2}+.....+\frac{1}{2n+n}\right]$
$=\displaystyle \lim_{n \to \infty}$$\left[\frac{1}{2+\frac{1}{n}}+\frac{1}{2+\frac{2}{n}}+...+\frac{1}{2+\frac{n}{n}}\right]$
$=\displaystyle \lim_{n \to \infty}$$\frac{1}{n}$$\displaystyle \sum_{n=1}^n$$\frac{1}{2+\frac{2}{n}}=$$\int\limits^{1}_0$$\frac{dx}{2+x}$
$=\left[log\left(2+x\right)]^{1}_{0}\right]=log\,3-log\,2=log \frac{3}{2}$