How to find $$ \lim_{n\rightarrow \infty} \frac {n^2}{n^3+n +1} + \frac {n^2}{n^3 +n +2} +.....+\frac {n^2}{n^3 + 2n}? $$
I was thinking about the Riemann sum, but I am not able to do it.
2026-04-29 20:23:18.1777494198
How to find $\lim_{n\rightarrow \infty} \frac {n^2}{n^3+n +1} + \frac {n^2}{n^3 +n +2} +.....+\frac {n^2}{n^3 + 2n}$?
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\begin{align*} \sum_{k=1}^{n}\dfrac{n^{2}}{n^{3}+n+k}\leq\sum_{k=1}^{n}\dfrac{n^{2}}{n^{3}+n+1}=\dfrac{n^{3}}{n^{3}+n+1} \end{align*} and \begin{align*} \sum_{k=1}^{n}\dfrac{n^{2}}{n^{3}+n+k}\geq\sum_{k=1}^{n}\dfrac{n^{2}}{n^{3}+n+n}=\dfrac{n^{3}}{n^{3}+2n}, \end{align*} now use Squeeze Theorem to conclude.