Let $c_n = \frac{1}{n^3} \sum_{i=1}^{n} i(1+\frac{i}{n})^2$ and find $\lim_{n \to\infty} c_n$ if it exists.
My idea was to show that $\lim_{n \to\infty} c_n = $ integral of something that converges, but I'm having a hard time representing the summation as an integral due to the $1/n^3$. If that was instead $1/n^2$ then it would be equal to $\int_{0}^{1} x(x+1)^2 \,dx$ (we create a N partitions equally spaced on [0,1], then represent the summation as an upper Riemann sum of $x(x+1)^2$ ) But I'm not sure how to do it with $1/n^3.$