Evaluating limit of a sum:$\lim_{n \to \infty}\frac1n\sum_{j=1}^n (a - {2j\over n})^3$ I am trying to solve this problem:
$$\lim_{n \to \infty}\sum_{j=1}^n {(a - {2j\over n})^3\over n}$$
My first instinct was to do a substition $x = {1 \over n}$ and rewrite the problem as:
$$\lim_{x \to 0}\sum_{j=1}^\infty {(a - {2jx})^3}x$$
Here it looks obvious that this is a Riemann sum that can be evaluated as an integral, but I'm stuck. I don't know how to calculate this as an integral. This is the best I could come up with:
$$\int_1^\infty {(a - {2jx})^3}dx$$
But I don't think it makes sense to evaluate at infinity for the upper bound. I think my integral is written wrong, but I'm not sure how to fix it. Any hints as to how to approach this problem properly would be appreciated.

Hint:
When $j$ changes from $1$ to $n$, then the term $\dfrac{j}{n}$ varies from $\dfrac{1}{n}$ to $1$, which leads to $0\leqslant x \leqslant 1.$
