Solve $\lim_{n \to \infty} \sum_{k=1}^{n}\frac{k^2}{2n^3 + k^3}$ using Riemann sums.

Question

I am familiar with the definition of Riemann sums and how they are used to evaluate definite integrals. But I am completely stuck with this expression. I can't seem to extract any patterns that relate to Riemann sums (other than the limit and summation term). Am I supposed to assume an interval in terms of $k$ and try to formulate an expression of the form $f(x_i^*)\Delta x$? Any hints would be greatly appreciated!

Answer 1

Hint

$$\sum_{k=1}^n\frac{k^2}{2n^3+k^3}=\frac{1}{n}\sum_{k=1}^n\frac{(k/n)^2}{2+(k/n)^3}.$$

Answer 2

It's $$\sum_{k=1}^n\frac{\left(\frac{k}{n}\right)^2}{2+\left(\frac{k}{n}\right)^3}\frac{1}{n}\rightarrow\int_0^1\frac{x^2}{2+x^3}dx=\frac{1}{3}\ln|2+x^3||_0^1=\ln\sqrt[3]{\frac{3}{2}}.$$