Question: If [x] denotes the greatest integer £ x, then ${\lim_{n \rightarrow \infty}}^{\frac{1}{n^{3}}}${[...

Question

If [x] denotes the greatest integer £ x, then

${\lim_{n \rightarrow \infty}}^{\frac{1}{n^{3}}}$ {[1²x] + [2²x] + [3²x] + ….+ [n²x]} equals :

Answer 1

Solution

xĪI, [x] = x

$\underset{n \rightarrow \infty}{Lim}\frac{1^{2}x + 2^{2}x + ..... + n^{2}x}{n^{3}}$ Ž $\underset{n \rightarrow \infty}{Lim}\frac{x(1^{2} + 2^{2} + ..... + n^{2})}{n^{3}}$

= x. $\frac{1}{3}$ (difference of power of N^r and N^r is one)

Question: If [x] denotes the greatest integer £ x, then \({\lim_{n \rightarrow \infty}}^{\frac{1}{n^{3}}}\){[...

Solution