First slide

Introduction to definite integration

Question

$lim_{n \to \infty} \frac{1}{n} [1 + \frac{n^{2}}{n^{2} + 1^{2}} + \frac{n^{2}}{n^{2} + 2^{2}} + \dots + \frac{n^{2}}{n^{2} + (n - 1)^{2}}]$
is equal to

Easy

A

$\frac{π}{2}$
B

$\frac{π}{3}$
C

$\frac{π}{4}$
D

$\frac{π}{6}$

Solution

$\begin{array}{l} lim_{n \to \infty} \frac{1}{n} [1 + \frac{n^{2}}{n^{2} + 1^{2}} + \frac{n^{2}}{n^{2} + 2^{2}} + \dots + \frac{n^{2}}{n^{2} + (n - 1)^{2}}] \\ = lim_{n \to \infty} \frac{1}{n} \sum_{r = 0}^{n - 1} \frac{n^{2}}{n^{2} + r^{2}} = lim_{n \to \infty} \frac{1}{n} \sum_{r = 0}^{n - 1} \frac{1}{1 + {(\frac{r}{n})}^{2}} \\ = \int_{0}^{1} \frac{d x}{1 + x^{2}} = {\tan^{- 1} x|}_{0}^{1} = \frac{π}{4} \end{array}$

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