Number Theory

    

A Pedagogical Spigot: A Binary Digit-Extracting Series for π from First Principles

Authors: Mar Detic

This paper presents a detailed derivation and analysis of the seriesπ =∞Xk=0 12k + 1 + 24k + 1 + 14k + 3 − 14k.We demonstrate its equivalence to a definite integral, prove its convergence to π via analytical evaluation,and analyze its convergence rate, showing that it yields approximately 3 correct decimal digits per5 terms. Crucially, we show that the factor of (−1/4)k imbues the series with the property of a spigotalgorithm for the binary digits of π. While not as computationally efficient as the Bailey—Borwein—Plouffeformula, this series’ derivation from a simple rational function using undergraduate-level techniquesoffers significant pedagogical value in understanding the construction of digit-extracting algorithms forfundamental constants.

Comments: 4 Pages.

Download: PDF

Submission history

[v1] 2024-11-26 13:29:46
[v2] 2024-12-19 02:35:22
[v3] 2024-12-27 21:14:23
[v4] 2025-05-06 20:54:56
[v5] 2025-08-29 23:10:56

Unique-IP document downloads: 1013 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus