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

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 th root.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsIterative Methods for Nonlinear Equations · Matrix Theory and Algorithms · Mathematical functions and polynomials
