Cancellations in power series of sine type

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 $$f(t):={\int }_0^t\frac{dx}{x}{\int }_0^x\frac{dy}{y}{\int }_0^y\frac{dz}{z}\{\sin x+\sin (x-y)-\sin (x-z)-\sin (x-y+z)\}.$$ We will derive various different representations of $f(t)$ by means of which it will be shown that ${\lim }_{t\to +\infty }f(t)=0$, disproving a conjecture by Z. Silagadze, claiming that this limit equals $-{\pi }^3/12$.

12 pages