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 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 ${\pi }^9\mathrm{/}{e}^8\approx 10$.The role of Archimede’s approximation $\pi =22\mathrm{/}7$ is discussed.