A Note on The Positivity of the Coefficients of Some Power Series Expansions

TL;DR

This paper proves a general result ensuring the positivity of coefficients in certain power series and applies it to confirm the positive sign of residual errors in Halley's iterations for the pth root.

Contribution

It introduces a new general theorem on power series coefficient positivity and applies it to a specific problem in iterative root-finding methods.

Findings

Established conditions for positivity of power series coefficients.

Confirmed the positive sign of residual errors in Halley's method for the pth root.

Provided a positive answer to Guo's 2010 question.

Abstract

In this short note, a general result concerning the positivity, under some conditions, of the coefficients of a power series is proved. This allows us to answer positively a question raised by Guo (2010) about the sign of the coefficients of a power series relating the residual errors in Halley's iterations for the $p$th root.