Asymptotics of alternating harmonic series with attenuation

TL;DR

This paper derives the asymptotic behavior of an alternating harmonic series with exponential attenuation as the parameter grows large, revealing oscillatory decay dominated by a specific exponential term.

Contribution

It provides the first detailed asymptotic analysis of the series and its connection to the Fourier transform of a related function, uncovering oscillatory decay patterns.

Findings

Asymptotics of the series involve oscillations and exponential decay.

The decay rate is dominated by 8((2\u03c0 t)^{1/2}).

Fourier transform asymptotics of a related function are established.

Abstract

We find the asymptotics of the series $\sum_{n=1}^\infty (-1)^n n^{-1} \exp(-t/n)$ as $t\to+\infty$. The answer is an oscillating function of $t$ dominated by $\exp(-(2\pi t)^{1/2})$. The intermediate step is to find the asymptotics of the two-dimensional Fourier transform $\hat F(\xi)$ of the function $F(x)=(1+\exp(\|x\|^2))^{-1}$ as $\|\xi\|\to\infty$.