Question

$\underset{n\to \infty }{lim}\frac{1^3+2^3+3^3+...n^3}{{(n^2+1)}^2}$

Answer 1

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

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

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

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

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

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

$n\to \infty$

$\frac{1}{n}\to 0\frac{1}{n^2}\to 0\frac{1}{n^3}\to 0\frac{1}{n^4}\to 0$

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

$=\frac{1}{4}$