Asymptotic expansion of a function defined by power series

TL;DR

This paper establishes conditions under which functions defined by Taylor series have asymptotic expansions in negative powers, providing formulas for the coefficients and an example with the arctangent function.

Contribution

It introduces a sufficient condition for the existence of asymptotic expansions in negative powers for functions defined by Taylor series and offers explicit formulas for the coefficients.

Findings

Derived formulas for asymptotic expansion coefficients

Provided a computational scheme for the arctangent function

Established a sufficient condition for the existence of such expansions

Abstract

We present a sufficient condition of existence of asymptotic expansion in negative power series for a function defined by Taylor series and unitary formulas for coefficients of this expansion. An example of computing scheme for arctangent function is represented.