The question states: Find the limit of the sequence
{$\displaystyle n^{2/3}[(n+1)^{1/3}-n^{1/3}]$}
I'm assuming they are asking as n approaches infinity, would anyone be able to offer tips on where to begin to solve?
$\displaystyle \lim_{n\to\infty}\sqrt[3]{n^2}\left(\sqrt[3]{n+1}-\sqrt[3]{n}\right)=$
$\displaystyle =\lim_{n\to\infty}\frac{\sqrt[3]{n^2}}{\sqrt[3]{(n+1)^2}+\sqrt[3]{n^2+n}+\sqrt[3]{n^2}}=$
$\displaystyle \lim_{n\to\infty}\frac{\sqrt[3]{n^2}}{\sqrt[3]{n^2}\left(\sqrt[3]{\left(1+\frac{1}{n}\right)^2}+\sqrt[3]{1+\frac{1}{n}}+1\right)}=$
$\displaystyle =\lim_{n\to\infty}\frac{1}{\sqrt[3]{\left(1+\frac{1}{n}\right)^2}+\sqrt[3]{1+\frac{1}{n}}+1}=\frac{1}{3}$
Thanks guys!
one step I don't seem to fully understand is going from:
$\displaystyle \lim_{n\to\infty}\sqrt[3]{n^2}\left(\sqrt[3]{n+1}-\sqrt[3]{n}\right)$
to
$\displaystyle \lim_{n\to\infty}\frac{\sqrt[3]{n^2}}{\sqrt[3]{(n+1)^2}+\sqrt[3]{n^2+n}+\sqrt[3]{n^2}}$
why does part of the factored expression all move into the denominator?
$\displaystyle \underset{n\to \infty }{\mathop{\lim }}\,\left( \sqrt[3]{n^{3}+n^{2}}-n \right),$ as i said before, put $\displaystyle t=\frac1n$ and the limit becomes $\displaystyle \underset{t\to 0}{\mathop{\lim }}\,\frac{\sqrt[3]{t+1}-1}{t},$ and then put $\displaystyle u=\sqrt[3]{t+1}$ and the limit is $\displaystyle \underset{u\to 1}{\mathop{\lim }}\,\frac{u-1}{u^{3}-1}=\underset{u\to 1}{\mathop{\lim }}\,\frac{1}{u^{2}+u+1}=\frac{1}{3}.$