I have a question regarding a sum.
Is the following expression finite and can be calculated?
$$\lim_{a\to\infty}\frac{1}{a}\sum_{b=0}^a \left(\frac{b}{a}\right)^2$$
Could I also approximate the sum by an integral since the upper index grows to infinity like
$$\lim_{a\to\infty}\int_{b=0}^a~db \frac{b^2}{a^3}$$ which would be $<\infty$?