How to evaluate a sum which contains limit variables? 
How to evaluate a sum which contains limit variables?

For example: $$\lim_{n\to\infty}\sum_{i=1}^n\frac{n-1}n\frac{1+i(n-1)}n $$
And would the result necessarily be rational, because each term appears to be the multiplication of two rational fractions? 

\begin{align}
S_n &= \frac{n-1}{n}\left(\frac{1}{n} n + \frac{n-1}{n} \frac{n(n+1)}{2}\right)\\
&= \frac{n-1}{n}\left( 1 + \frac{n^2-1}{2}\right)
\end{align}
This seems to go towards $\infty$.
