Answer
$$\sum_{i=1}^{n} \frac{2 i^{3}-3 i}{n^{4}} =\frac{n^3+2n^2-2n-3}{2n^3}$$
$\begin{array}{|c|c|c|c|c|}\hline 10 & {100} & {1000} & {10000} \\ \hline 0.5885 & {0.0509895} & { 0.500999 } & {0.50009999} \\ \hline\end{array}$
Work Step by Step
Since
\begin{align*}
\sum_{i=1}^{n} \frac{2 i^{3}-3 i}{n^{4}}&=\frac{1}{n^4}\sum_{i=1}^{n} (2 i^{3}-3 i)\ \ \text{Use } ( \sum_{i=1}^{n} k a_{i}=k \sum_{i=1}^{n} a_{i})\\
&=\frac{1}{n^4}\left(2\sum_{i=1}^{n} i^{3}-3 \sum_{i=1}^{n} i \right)\ \ \text{Use } (\sum_{i=1}^{n}\left(a_{i} \pm b_{i}\right)=\sum_{i=1}^{n} a_{i} \pm \sum_{i=1}^{n} b_{i})\\
&=\frac{1}{n^4}\left( \frac{n^2(n+1)^2}{2}-\frac{3n(n+1)}{2}\right)\\
&=\frac{1}{n^4}\left(\frac{n^4+2n^3-2n^2-3n}{2}\right)\\
&=\frac{n^3+2n^2-2n-3}{2n^3}
\end{align*}
and
$\begin{array}{|c|c|c|c|c|}\hline 10 & {100} & {1000} & {10000} \\ \hline 0.5885 & {0.0509895} & { 0.500999 } & {0.50009999} \\ \hline\end{array}$
