<s>[INST] infinite sum limit how to find the following Hi what is the limit of the following sum:
$$\lim \limits_{n\rightarrow\infty}\frac{2}{n^2}\sum\limits_{j=0}^{n-1}\sum\limits_{k=j+1}^{n-1}\frac{k}{n}$$
Thanks a lot!
 [/INST] Assuming the problem is correctly written, we have
$$\begin{align}
\lim_{n\rightarrow\infty}
\frac{2}{n^2}\sum_{j=0}^{n-1}\sum_{k=j+1}^{n-1}\frac{j}{n}
&=
\lim_{n\rightarrow\infty}
\frac{2}{n^3}\sum_{j=0}^{n-1}j\sum_{k=j+1}^{n-1}1 \\
&=
\lim_{n\rightarrow\infty}
\frac{2}{n^3}\sum_{j=0}^{n-1}j(n-j-1) \\
&=
\lim_{n\rightarrow\infty}
\frac{1}{n^3}\left\{
(n-1)\sum_{j=0}^{n-1}j-\sum_{j=0}^{n-1}j^2
\right\} \\
&=
\lim_{n\rightarrow\infty}\frac{1}{n^3}\left\{
(n-1)\frac{n(n-1)}{2}-\frac{n(n-1)(2n-1)}{6}
\right\} \\
&= \frac{1}{2}-\frac{1}{3} \\
&= \frac{1}{6}
\end{align}
$$
</s>