Relations between $e$ and $\ddot{I}€$: Nilakantha's series and Stirling's formula

Abstract

Approximate relations between $e$ and $\ddot{I}€$ are reviewed, some new connections being established. Nilakantha's series expansion for $\ddot{I}€$ 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 $e+2\ddot{I}€\approx 9$. Using Stirling's series enables us to illustrate the relations $\ddot{I}{€}^4+\ddot{I}{€}^5\approx e^6$ and $\ddot{I}{€}^9/e^8\approx 10$.The role of Archimede's approximation $\ddot{I}€=22/7$ is discussed.

6 pages, some explanations are extended