Calculate limit of a sum as an integral

Question

I need to calculate this limit as a definite integral but it doesn't look like Riemann sum at all:

$$ \lim_{x\to\infty} n^2 \sum_{i=1}^{n} \frac{1}{(n + i + 1)^3} $$

What would be an approach to this? Thanks in advance.

Answer 1

L = $ \lim_{n\to \infty} \sum_{i=0}^n\frac{n^2}{(n+i+1)^3}$ [taking n common from the denominator]

L = $ \lim_{n\to \infty} \frac{1}{n} \sum_{i=0}^n\frac{1}{(1+\frac{i}{n}+\frac{1}{n})^3}$ [converting to integral]

L = $\int_0^1 \frac{1}{(1+x)^3}\, dx$

L = $\frac{3}{8}$ = 0.375