Find a limit using Riemann Sums > Find $$\lim_{n\to \infty} \sum_{r=1}^n \frac {r^2}{n^3+r^2}$$ Now the limit is very easy to go using the sandwich theorem which yields in the answer as $\frac 13$. But I want to know if there is any corresponding solution which inturn uses Riemann Sums and Integrals to evaluate this limit. Edit: > Another such question is $$\lim_{n\to \infty} \sum_{r=1}^n \frac {n}{n^2+r}$$ It could have been much easier to solve if the above question had been $$\lim_{n\to \infty} \sum_{r=1}^n \frac {n}{n^2+r^2}$$ which simplifies to $$\int_0^1 \frac {dx}{1+x^2}=\frac {\pi}{4}$$

$$\sum_{r=1}^{n}\frac{r^2}{n^3+r^2}\leq \frac{1}{n}\sum_{r=1}^{n}\left(\frac{r}{n}\right)^2 \to \int_{0}^{1}x^2\,dx = \frac{1}{3} $$ and $$\sum_{r=1}^{n}\frac{r^2}{n^3+r^2}\geq \frac{n^3}{n^3+n^2}\sum_{r=1}^{n}\left(\frac{r}{n}\right)^2 \to \frac{1}{3}.$$ Essentially, the $r^2$ term in the denominator is negligible (with or without Riemann sums).
We cannot state the same for $$ \lim_{n\to +\infty}\sum_{r=1}^{n}\frac{r^2}{n^3+r^{\color{red}{3}}}=\lim_{n\to +\infty}\frac{1}{n}\sum_{r=1}^{n}\frac{\left(\frac{r}{n}\right)^2}{1+\left(\frac{r}{n}\right)^3}=\int_{0}^{1}\frac{x^2\,dx}{1+x^3}=\frac{\log 2}{3}. $$