Relations between e and π: Nilakantha's series and Stirling's formula

2022/06/13 by V. Yu. Irkhin, Irkhin, V. Yu.
Chemical Engineering · Decision Sciences · Mathematics · #Advanced Mathematical Identities #FOS: Mathematics #History and Overview (math.HO) #Scientific Measurement and Uncertainty Evaluation #Thermodynamic properties of mixtures

paper · pdf · doi:10.48550/arxiv.2206.07174

openalex publication_date 2022/06/13 · openalex created_date 2022/06/18 · openalex updated_date 2026/08/04

Abstract

Approximate relations between e and π are reviewed, some new connections being established. Nilakantha's series expansion for π is transformed to accelerate its convergence. Its comparison with the standard inverse-factorial expansion for e is performed to demonstrate similarity in several first terms. This comparison clarifies the origin of the approximate coincidence e+2π≈ 9. Using Stirling's series enables us to illustrate the relations π⁴+π⁵ ≈ e⁶ and π⁹/e⁸ ≈ 10.The role of Archimede's approximation π=22/7 is discussed.

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