<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] You can proceed from here as $$\lim_{n\to \infty}\sum\limits_{k=0}^n \frac{k^2}{k^3+n^3}= \lim_{n\to\infty}\sum\limits_{k=0}^n \;\frac{1}{n}\frac{\frac{k^2}{n^2}}{\frac{k^3}{n^3}+1}= \lim_{h\to 0} \;h\sum_{k=1}^n \frac{{(kh)}^2}{1 + {(kh)}^3}= \int_0^1 \frac{x^2}{1+x^3} \mathrm{d}x\;.$$
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