Monotonicity of Coefficients of Reciprocal Power Series

Abstract

Given a power series $f(x)=\sum_{n=1}^\infty\,a_n\,x^n$ with nonnegative coefficients satisfying $\sum_{n=1}^\infty\,a_n=1$ we give sufficient conditions on the sequence $(a_n)$ to guarantee that the coefficients of the Taylor series of $h(x)=1/(1-f(x))$ form a nonincreasing sequence. This type of result is useful when one wishes to apply Tauberian theorems.