<s>[INST] Find the limit of $\lim_{n\to \infty}n^2({1\over{n^3+1^3}}+{1\over{n^3+2^3}}+\cdots+{1\over{n^3+n^3}}).$ Find the limit of $$\lim_{n\to \infty}n^2({1\over{n^3+1^3}}+{1\over{n^3+2^3}}+\cdots+{1\over{n^3+n^3}}).$$
I'm not sure how to evaluate this limit. Any hints or solutions are greatly appreciated. I turned this into $\int_o^1{1\over {1+x^3}}dx$ using algebra. Is this correct or is there a better way?
 [/INST] Your integral is correct, since the terms of your sum can be rewritten as $$\frac{n^2}{n^3+i^3} = \frac{1}{n} \cdot \frac{1}{1+(i/n)^3}$$, which means that your sum is a Riemann sum for the corresponding integral.
If you are having a hard time getting started on evaluating the integral, you can factor $x^3+1 = (x+1)(x^2-x+1)$, and then focus on rewriting your integrand in the following form (using the method of partial fractions)
$$
\frac{1}{x^3+1} = \frac{A}{x+1} + \frac{B(2x-1)}{x^2-x+1} +\frac{C}{x^2-x+1}
$$
(also note that $x^2-x+1 = \left(x-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2$.)
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