What is the limit of the multidimensional integral?

Question

What is the limit of the integral $$\int_{[0,1]^n}\frac{x_1^5+x_2^5 + \cdots +x_n^5}{x_1^4+x_2^4 + \cdots +x_n^4} \, dx_1 \, dx_2 \cdots dx_n$$ as $n \to \infty ?$

Answer 1

Here is a heuristics: Let $U_i$ be i.i.d. random variables having the uniform distribution on $[0, 1]$. Then the integral is equal to

$$ \Bbb{E} \left[ \frac{U_{1}^{5} + \cdots + U_{n}^{5}}{U_{1}^{4} + \cdots + U_{n}^{4}} \right] $$

Now, from the strong law of large number we find that

$$ \frac{U_{1}^{5} + \cdots + U_{n}^{5}}{U_{1}^{4} + \cdots + U_{n}^{4}} \to \frac{\Bbb{E} U_{1}^{5}}{\Bbb{E} U_{1}^{4}} = \frac{5}{6} \quad \text{a.s.} $$

Therefore the limit of the integral is $\frac{5}{6}$.

The only unjustified part of this argument is that we intercanged the order of the expectation and the limit. This can be justified by the dominated convergence theorem, together with the observation that

$$ \frac{U_{1}^{5} + \cdots + U_{n}^{5}}{U_{1}^{4} + \cdots + U_{n}^{4}} \leq 1. $$