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

TL;DR

This paper explores mathematical relations between e and pi, transforming series to accelerate convergence, comparing expansions, and establishing new approximate identities involving these fundamental constants.

Contribution

It introduces new connections between e and pi, accelerates convergence of Nilakantha's series, and clarifies the origin of certain approximate identities.

Findings

Nilakantha's series transformed for faster convergence

Comparison reveals similarity in initial terms of series expansions

New approximate identities involving e and pi established

Abstract

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.