First slide

Introduction to definite integration

Question

$\underset{n \to \infty}{L t} [\frac{1^{m} + 2^{m} + 3^{m} + ..... + n^{m}}{n^{m + 1}}]$

Moderate

A

$\frac{1}{m + 1}$
B

$\frac{1}{m + 2}$
C

$\frac{1}{m}$
D

$\frac{1}{m + 3}$

Solution

$\begin{array}{l} \lim_{n \to \infty} \frac{1}{n} [\frac{1^{m}}{n^{m}} + \frac{2^{m}}{n^{m}} + \frac{3^{m}}{n^{m}} + \dots . + \frac{n^{m}}{n^{m}}] \\ = \lim_{n \to \infty} \begin{array}{l} \frac{1}{n} \sum_{r = 1}^{n} {(\frac{r}{n})}^{m} = \int_{0}^{1} x^{m} d x = {(\frac{x^{m + 1}}{m + 1})}_{0}^{1} = \frac{1}{m + 1} \end{array} \end{array}$

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