Relations between \(e\) and \(\pi\): Nilakantha's series and Stirling's formula

Approximate relations between \(e\) and \(\pi\) are reviewed, some new connections being established. Nilakantha's series expansion for \(\pi\) is transformed to accelerate its convergence. Its comparison with the standard inverse-factorial expansion for \(e\) is performed to demonstrate simila...

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Bibliographic Details
Published in:arXiv.org 2022-09
Main Author: V Yu Irkhin
Format: Article
Language:English
Subjects:
Series expansion
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Summary:Approximate relations between \(e\) and \(\pi\) are reviewed, some new connections being established. Nilakantha's series expansion for \(\pi\) 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 \(e+2\pi \approx 9\). Using Stirling's series enables us to illustrate the relations \(\pi^4+\pi^5 \approx e^6\) and \(\pi^{9}/e^{8} \approx 10\).The role of Archimede's approximation \(\pi=22/7\) is discussed.
ISSN:2331-8422