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arXiv · 1802.09214

An Asymptotic Series for an Integral

Michael E. Hoffman·Markus Kuba·Moti Levy·Guy Louchard
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Abstract

We obtain an asymptotic series $\sum_{j=0}^\infty\frac{I_j}{n^j}$ for the integral $\int_0^1[x^n+(1-x)^n]^{\frac1{n}}dx$ as $n\to\infty$, 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 $\Vert(U,1-U)\Vert_n$ as $n$ tends to infinity and we also discuss $\Vert(U_1,U_2,\dots,U_r)\Vert_n$ for standard uniformly distributed random variables.

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Michael E. Hoffman, Markus Kuba, Moti Levy, Guy Louchard. 2018-02-26. An Asymptotic Series for an Integral. https://arxiv.org/abs/1802.09214

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