Relations between $e$ and $π$: Nilakantha's series and Stirling's formula
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866913762849587200 |
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| author | Irkhin, V. Yu. |
| author_facet | Irkhin, V. Yu. |
| contents | 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 $e+2π\approx 9$. Using Stirling's series enables us to illustrate the relations $π^4+π^5 \approx e^6$ and $π^{9}/e^{8} \approx 10$.The role of Archimede's approximation $π=22/7$ is discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_07174 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Relations between $e$ and $π$: Nilakantha's series and Stirling's formula Irkhin, V. Yu. History and Overview 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 $e+2π\approx 9$. Using Stirling's series enables us to illustrate the relations $π^4+π^5 \approx e^6$ and $π^{9}/e^{8} \approx 10$.The role of Archimede's approximation $π=22/7$ is discussed. |
| title | Relations between $e$ and $π$: Nilakantha's series and Stirling's formula |
| topic | History and Overview |
| url | https://arxiv.org/abs/2206.07174 |