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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.15803 |
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| _version_ | 1866915719160004608 |
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| author | Ern, Thang Pang Wangsa, Devandhira Wijaya |
| author_facet | Ern, Thang Pang Wangsa, Devandhira Wijaya |
| contents | In 1914, Ramanujan presented a collection of 17 elegant and rapidly converging formulae for $π$. Among these, one of the most celebrated is the following series: \[\frac{1}π=\frac{2\sqrt{2}}{9801}\sum_{n=0}^{\infty}\frac{26390n+1103}{\left(n!\right)^4} \frac{\left(4n\right)!}{396^{4n}}\] In this paper, we give a full proof of this classic formula using hypergeometric series and a special type of lattice sums due to Zucker and Robertson. We will also use some results by Dirichlet and Edwards in algebraic number theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Proof of Ramanujan's Classic $π$ Formula Ern, Thang Pang Wangsa, Devandhira Wijaya Number Theory In 1914, Ramanujan presented a collection of 17 elegant and rapidly converging formulae for $π$. Among these, one of the most celebrated is the following series: \[\frac{1}π=\frac{2\sqrt{2}}{9801}\sum_{n=0}^{\infty}\frac{26390n+1103}{\left(n!\right)^4} \frac{\left(4n\right)!}{396^{4n}}\] In this paper, we give a full proof of this classic formula using hypergeometric series and a special type of lattice sums due to Zucker and Robertson. We will also use some results by Dirichlet and Edwards in algebraic number theory. |
| title | A Proof of Ramanujan's Classic $π$ Formula |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.15803 |