First slide

Introduction to definite integration

Question

$T h e v a l u e o f \lim_{n \to \infty} n [\frac{1}{(n + 1) (n + 2)} + \frac{1}{(n + 2) (n + 4)} + \dots + \frac{1}{6 n^{2}}]$

Moderate

A

$\log 3$
B

log2
C

$\log (\frac{3}{2})$
D

$\log (\frac{2}{3})$

Solution

$The given limit is$
$\begin{array}{rcl} L & = & lim_{n \to \infty} \sum_{r = 1}^{n} n \cdot \frac{1}{(n + r) (n + 2 r)} \\ = & lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{n} \frac{1}{(1 + \frac{r}{n}) (1 + \frac{2 r}{n})} \\ = & \int_{0}^{1} \frac{d x}{(1 + x) (1 + 2 x)} \\ = & \int_{0}^{1} (\frac{- 1}{1 + x} + \frac{2}{1 + 2 x}) d x \\ = & [- \log (1 + x) + \log (1 + 2 x)]_{0}^{1} \\ = & [(- \log 2 + \log 3) - (- \log 1 + \log 1)] \\ = & \log (\frac{3}{2}) \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App