On the coefficients in an asymptotic expansion of (1+1/x)^x

2021/05/08 by T. M. Dunster, Dunster, T. M., Jessica M. Perez +1
Mathematics · #30E15 #30E20 #41A58 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:30E15 #msc:30E20 #msc:41A58

paper · pdf · doi:10.48550/arxiv.2105.03794

arxiv created 2021/08/10 · arxiv updated 2021/08/11

Abstract

The function g(x)= (1+1/x)^x has the well-known limit e as x→∞. The coefficients c_j in an asymptotic expansion for g(x) are considered. A simple recursion formula is derived, and then using Cauchy's integral formula the coefficients are approximated for large j. From this it is shown that |c_j|→1 as j→∞.

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