An Asymptotic Series for an Integral

We obtain an asymptotic series ${\sum }_{j=0}^{\mathrm{\infty }}\frac{I_j}{n^j}$ for the integral ${\int }_0^1[x^n+(1-x{)}^n{]}^{\frac{1}{n}}dx$ as $n\to \mathrm{\infty }$, and compute $I_j$ in terms of alternating (or “colored”) multiple zeta value. We also show that $I_j$ is a rational polynomial the ordinary zeta values, and give explicit formulas for $j\le 12$. As a byproduct, we obtain precise results about the convergence of norms of random variables and their moments. We study $\mathrm{\parallel }(U,1-U){\mathrm{\parallel }}_n$ as $n$ tends to infinity and we also discuss $\mathrm{\parallel }(U_1,U_2,\dots ,U_r){\mathrm{\parallel }}_n$ for standard uniformly distributed random variables.