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Nested integrals and limit

Posted: Thu Nov 12, 2015 3:15 pm
by Tolaso J Kos
I am posting this exercise here but perhaps it can fit elsewhere.

Show that the integral:

$$\begin{eqnarray*}V_n=\int_0^1 \int_0^1 \cdots \int_0^1 \frac{x_1^2+x_2^2 +\cdots +x_n^2}{x_1+x_2+\cdots+x_n}\, {\rm d}x_1 \, {\rm d} x_2 \cdots \,{\rm d}x_n \end{eqnarray*}$$

converges to $2/3$ as $n \rightarrow +\infty$ and that the product $n(V_n -2/3)$ remains bounded.
Source
American Mathematical Monthly problem 3408 (Mar. 1932).