<s>[INST] To evaluate the limits $\lim\limits_{n \to \infty} \{\frac{1}{1+n^3}+\frac{4}{8+n^3}+\ldots +\frac{n^2}{n^3+n^3}\}$ To me it seems like that we need to manipulate the given sum into the Riemann sum of some function. First writing in the standard summation form;
$$\{\frac{1}{1+n^3}+\frac{4}{8+n^3}+\ldots +\frac{n^2}{n^3+n^3}\}=\sum\limits_{k=0}^n \frac{k^2}{k^3+n^3}$$
I have tried factoring out $\frac{1}{n}$ from the above sum. But I cant seem to get into the right form. Any help!
 [/INST] Note that
$$\frac{k^2}{k^3+n^3}=\dfrac1n\cdot\dfrac{\left( \frac{k}{n} \right)^2}{\left( \frac{k}{n} \right)^3+1}$$
so you can rewrite the summation to be
$$\begin{align}\lim_{n\to \infty} \Big(\frac{1}{1+n^3}+\frac{4}{8+n^3}+\ldots +\frac{n^2}{n^3+n^3}\Big) &= \lim_{n\to \infty} \sum\limits_{k=0}^n \frac{k^2}{k^3+n^3}\\
&= \lim_{n\to \infty} \sum\limits_{k=0}^n \dfrac1n\cdot\dfrac{\left( \frac{k}{n} \right)^2}{\left( \frac{k}{n} \right)^3+1}\\
&= \lim_{h\to 0} \;h\sum_{k=1}^n \frac{{(kh)}^2}{1 + {(kh)}^3}\\
&= \int_0^1 \frac{x^2}{1+x^3} \mathrm{d}x\;\\
&= \frac{1}{3} \ln(1+x^3) \Big|_0^1\\
&=  \frac{1}{3} \ln 2
\end{align}$$
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