On the irrationality of certain super-polynomially decaying series
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911248380067840 |
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| author | Crmarić, Tonći Kovač, Vjekoslav |
| author_facet | Crmarić, Tonći Kovač, Vjekoslav |
| contents | We give a negative answer to a question by Paul Erdős and Ronald Graham on whether the series \[ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)\cdots(n+f(n))} \] has an irrational sum whenever $(f(n))_{n=1}^{\infty}$ is a sequence of positive integers converging to infinity. To achieve this, we generalize a classical observation of Sōichi Kakeya on the set of all subsums of a convergent positive series. We also discuss why the same problem is likely difficult when $(f(n))_{n=1}^{\infty}$ is additionally assumed to be increasing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the irrationality of certain super-polynomially decaying series Crmarić, Tonći Kovač, Vjekoslav Number Theory Classical Analysis and ODEs We give a negative answer to a question by Paul Erdős and Ronald Graham on whether the series \[ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)\cdots(n+f(n))} \] has an irrational sum whenever $(f(n))_{n=1}^{\infty}$ is a sequence of positive integers converging to infinity. To achieve this, we generalize a classical observation of Sōichi Kakeya on the set of all subsums of a convergent positive series. We also discuss why the same problem is likely difficult when $(f(n))_{n=1}^{\infty}$ is additionally assumed to be increasing. |
| title | On the irrationality of certain super-polynomially decaying series |
| topic | Number Theory Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2504.18712 |