Question

${lim}_{n\to \infty }\frac{1^3+2^3+3^3+....+n^3}{n^4}$

Answer 1

${lim}_{n\to \infty }\frac{1^3+2^3+3^3+....+n^3}{n^4}$

$={lim}_{n\to \infty }\frac{{\left[\frac{1}{2}n\right(n+1\left)\right]}^2}{n^4}$

$[1^3+2^3+3^3+....+n^3={\left(\frac{1}{2}n\right(n+1\left)\right)}^2]$

$={lim}_{n\to \infty }\frac{\frac{1}{4}{n}^2(n+1{)}^2}{n^4}$

$={lim}_{n\to \infty }\frac{\frac{1}{4}{n}^2(n^2+1+2n)}{n^4}$

$={lim}_{4n\to \infty }\frac{n^4(1+\frac{1}{n^2}+\frac{2}{n})}{n^4}\left[\frac{\infty }{\infty }form\right]$

$=\frac{1}{4}{lim}_{n\to \infty }\frac{(1+\frac{1}{n^2}+\frac{2}{n})}{1}$

$Whenn\to \infty ,then\frac{1}{n}\to 0=\frac{1}{4}$