A simple computation of $ζ(2k)$ by using Bernoulli polynomials and a telescoping series
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2012
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| _version_ | 1866929654938468352 |
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| author | Ciaurri, Ó. Navas, L. M. Ruiz, F. J. Varona, J. L. |
| author_facet | Ciaurri, Ó. Navas, L. M. Ruiz, F. J. Varona, J. L. |
| contents | We present a new proof of Euler's formulas for $ζ(2k)$, where $k = 1,2,3,...$, which uses only the defining properties of the Bernoulli polynomials, obtaining the value of $ζ(2k)$ by summing a telescoping series. Only basic techniques from Calculus are needed to carry out the computation. The method also applies to $ζ(2k+1)$ and the harmonic numbers, yielding integral formulas for these. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1209_5030 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | A simple computation of $ζ(2k)$ by using Bernoulli polynomials and a telescoping series Ciaurri, Ó. Navas, L. M. Ruiz, F. J. Varona, J. L. Number Theory 40C15, 11M06 We present a new proof of Euler's formulas for $ζ(2k)$, where $k = 1,2,3,...$, which uses only the defining properties of the Bernoulli polynomials, obtaining the value of $ζ(2k)$ by summing a telescoping series. Only basic techniques from Calculus are needed to carry out the computation. The method also applies to $ζ(2k+1)$ and the harmonic numbers, yielding integral formulas for these. |
| title | A simple computation of $ζ(2k)$ by using Bernoulli polynomials and a telescoping series |
| topic | Number Theory 40C15, 11M06 |
| url | https://arxiv.org/abs/1209.5030 |