Cancellations in power series of sine type

2015/05/03 by Juan Arias de Reyna, de Reyna, Juan Arias · 1 citation
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1505.00440

Abstract

We present a method to study the behavior of a power series of type f(x):=∑_n=0^∞ (-1)ⁿ c_n\fracx²ⁿ⁺¹(2n+1)! when x→∞. We apply our method to study the function f(t):=∫₀^t(dx)/(x)∫₀^x(dy)/(y)∫₀^y(dz)/(z)\ sin x+sin(x-y)-sin(x-z)-sin(x-y+z)\. We will derive various different representations of f(t) by means of which it will be shown that lim_t→+∞f(t)=0, disproving a conjecture by Z. Silagadze, claiming that this limit equals -π³/12.

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