First slide

Introduction to definite integration

Question

$E v a l u a t e lim_{n \to \infty} \frac{(1^{2} + 2^{2} + 3^{3} + \dots + n^{2}) (1^{3} + 2^{3} + 3^{3} + \dots + n^{3})}{1^{6} + 2^{6} + 3^{6} + \dots + n^{6}}$

Moderate

A

$7$
B

$12$
C

$\frac{7}{12}$
D

$\frac{12}{7}$

Solution

$\begin{array}{rcl} T h e g i v e n l i m i t i s \\ lim_{n \to \infty} \frac{\sum_{r = 1}^{n} r^{2} \times \sum_{r = 1}^{n} r^{3}}{\sum_{r = 1}^{n} r^{6}} & = & lim_{n \to \infty} \frac{n^{2} \sum_{r = 1}^{n} {(\frac{r}{n})}^{2} \times n^{3} \sum_{r = 1}^{n} {(\frac{r}{n})}^{3}}{n^{6} \sum_{r = 1}^{n} {(\frac{r}{n})}^{6}} \\ = & lim_{n \to \infty} \frac{\frac{1}{n} \sum_{r = 1}^{n} {(\frac{r}{n})}^{2} \times \frac{1}{n} \sum_{r = 1}^{n} {(\frac{r}{n})}^{3}}{\frac{1}{n} \sum_{r = 1}^{n} {(\frac{r}{n})}^{6}} \\ = & \frac{\int_{0}^{1} x^{2} d x \int_{0}^{1} x^{3} d x}{\int_{0}^{1} x^{6} d x} = \frac{{[\frac{x^{3}}{3}]}_{0}^{1} {[\frac{x^{4}}{4}]}_{0}^{1}}{{[\frac{x^{7}}{7}]}_{0}^{1}} = \frac{\frac{1}{3} \times \frac{1}{4}}{\frac{1}{7}} = \frac{7}{12} \end{array}$

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