First slide

Introduction to definite integration

Question

The value of
$lim_{n \to \infty} (\frac{1}{n} + \frac{n}{(n + 1)^{2}} + \frac{n}{(n + 2)^{2}} + \dots + \frac{n}{(2 n - 1)^{2}})$ is

Easy

A

1
B

$1 / 3$
C

$1 / 2$
D

$3 / 2$

Solution

Required $limit - lim_{n \to \infty} \sum_{r = 0}^{n - 1} \frac{n}{(n + r)^{2}}$
$\begin{array}{l} = lim_{n \to \infty} \frac{1}{n} \sum_{r = 0}^{n - 1} \frac{1}{{(1 + \frac{r}{n})}^{2}} \\ = \int_{0}^{1} \frac{d x}{(1 + x)^{2}} = - {\frac{1}{1 + x}|}_{0}^{1} \\ = - \frac{1}{2} + 1 = \frac{1}{2} \end{array}$

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