limn→∞1n3+22n3+32n3+...+n2n3

limn→∞1n3+22n3+32n3+..........+n2n3 Given limit written as =limn→∞∑r=1n r2n3 =limn→∞ 1n∑r=1n r2n2 =∫01x2dx =x3301 =13

limn→∞1n3+22n3+32n3+...+n2n3

limn→∞1n3+22n3+32n3+..........+n2n3 Given limit written as =limn→∞∑r=1n r2n3 =limn→∞ 1n∑r=1n r2n2 =∫01x2dx =x3301 =13

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$\lim_{n \to \infty} [\frac{1}{n^{3}} + \frac{2^{2}}{n^{3}} + \frac{3^{2}}{n^{3}} + . . . + \frac{n^{2}}{n^{3}}]$

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$\frac{1}{2}$

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{3}{2}$

answer is B.

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Detailed Solution

$\begin{array}{rcl} \lim_{n \to \infty} [\frac{1}{n^{3}} + \frac{2^{2}}{n^{3}} + \frac{3^{2}}{n^{3}} + . . . . . . . . . . + \frac{n^{2}}{n^{3}}] \\ G i v e n l i m i t w r i t t e n a s \\ = & \lim_{n \to \infty} \sum_{r = 1}^{n} \frac{r^{2}}{n^{3}} \\ = & \lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{n} \frac{r^{2}}{n^{2}} \\ = & \int_{0}^{1} x^{2} d x \\ = & {(\frac{x^{3}}{3})}_{0}^{1} \\ = & \frac{1}{3} \end{array}$