query on limit of a function What is the limit when $n \to \infty$?
$$\lim_{n \to \infty} \frac{1}{n^4} \sum_{J=0}^{2n-1} J^3=?$$
 A: Hint: $$1^3+2^3+3^3+\cdots+k^3=\left(\frac{k(k+1)}{2}\right)^2.$$
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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\begin{align}
\lim_{n \to \infty}\bracks{{1 \over n^{4}}\sum_{J = 0}^{2n - 1}J^{3}} &\ =\
\overbrace{\lim_{n \to \infty}\bracks{{1 \over \pars{n + 1}^{4} - n^{4}}
\pars{\sum_{J = 0}^{2n + 1}J^{3} - \sum_{J = 0}^{2n - 1}J^{3}}}}
^{\ds{\mbox{Stolz-Ces$\mrm{\grave{a}}$ro Theorem}}}
\\[5mm] & =
\lim_{n \to \infty}\,\,
{\pars{2n + 1}^{3} + \pars{2n}^{3} \over 4n^{3} + 6n^{2} + 4n + 1} =
\lim_{n \to \infty}\,\,
{16n^{3} + 12n^{2} + 6n + 1\over 4n^{3} + 6n^{2} + 4n + 1} = \bbx{\ds{4}}
\end{align}
