Cancellations in power series of sine type

TL;DR

This paper introduces a new method to analyze the asymptotic behavior of sine-type power series and applies it to a complex integral function, ultimately disproving a prior conjecture about its limit at infinity.

Contribution

The paper develops a novel approach for studying sine-type power series behavior at infinity and provides a counterexample to a conjecture regarding the limit of a specific integral function.

Findings

The limit of the studied function as t approaches infinity is zero.

The paper presents multiple representations of the function f(t).

It disproves Silagadze's conjecture that the limit equals -π^3/12.

Abstract

We present a method to study the behavior of a power series of type $$f(x):=\sum_{n=0}^\infty (-1)^n c_n\frac{x^{2n+1}}{(2n+1)!}$$ when $x\to\infty$. We apply our method to study the function $$f(t):=\int_0^t\frac{dx}{x}\int_0^x\frac{dy}{y}\int_0^y\frac{dz}{z}\bigl\{ \sin x+\sin(x-y)-\sin(x-z)-\sin(x-y+z)\bigr\}.$$ We will derive various different representations of $f(t)$ by means of which it will be shown that $\lim_{t\to+\infty}f(t)=0$, disproving a conjecture by Z. Silagadze, claiming that this limit equals $-\pi^3/12$.