Compute $\lim_{n\to\infty}\int_{0}^{1}\int_{0}^{1}...\int_{0}^{1} \frac{x_1^2+x_2^2+ ... +x_n^2 }{x_1+x_2+ ... +x_n} \space dx_1dx_2...dx_n$
Convergence in probability: $\lim_{n\to\infty}\int_0^1\cdots\int_0^1\frac{x_1^2+x_2^2+\cdots+x_n^2}{x_1+x_2+\cdots +x_n}dx_1\cdots dx_n=\frac23$