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? 

The general approach to evaluating such limits of sums is as follows. Say we are considering
$$
\lim_{n\to\infty}\sum_{i=1}^{f(n)}g(n)\approx \lim_{n\to\infty}\int_0^{f(n)}g(x)\ dx.
$$
For any reasonable functions $f$ and $g$, you can make the $\approx$ rigorous by using Riemann sums for the integral. Then we use something called the characteristic function:
$$
\lim_{n\to\infty}\int_0^{f(n)}g(x)\ dx=\lim_{n\to\infty}\int_0^{\infty}\chi_{[0,f(n)]}g(x)\ dx.
$$
Say that $\lim_{n\to\infty}f(n)=L$. Then we can pass the limit inside the integral to obtain
$$
\lim_{n\to\infty}\sum_{i=1}^{f(n)}g(n)\approx \int_0^{\infty}\chi_{[0,L]}g(x)\ dx.
$$
Note that there are some more hypotheses on $g$ necessary in order to pass the limit inside. The resulting integral can be computed using integration by parts if necessary (for instance, if there isn't a closed form antiderivative for $g(x)$).
If you have other specific examples, I could provide a more concrete answer.
