Question

Value of the following expression is $\displaystyle \lim_{n \to \infty}$ $\frac{1}{n^{3}}\left(1^{2}+2^{2}+3^{2}+ ..... +n^{2}\right)$

• A. $\frac{1}{3}$
• B. $\frac{1}{6}$
• C. $\frac{1}{2}$
• D. $\frac{2}{3}$

Answer 1

Answer: (A)

$\frac{1}{3}$

Answer 2

$\displaystyle \lim _{n \rightarrow \infty} \frac{1}{n^{3}}\left(1^{2}+2^{2}+3^{2}+\ldots +n^{2}\right)$
$=\displaystyle \lim _{n \rightarrow \infty} \frac{1}{n^{3}}\left[\frac{n(n+1)(2 n+1)}{6}\right]$
$=\displaystyle \lim _{n \rightarrow \infty} \frac{\left(1+\frac{1}{n}\right)\left(2+\frac{1}{n}\right)}{6}$
$=\frac{1 \times 2}{6}=\frac{1}{3}$