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.