Rapidly Converging Series for ζ(2n+1) from Fourier Series
Ever since Euler first evaluated ζ(2) and ζ(2m), numerous interesting solutions of the problem of evaluating the ζ(2m) (m∈ℕ) have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation of ζ(2m) (m∈ℕ) is known for ζ(2m+1) (m∈ℕ) or even for any special case...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/457620 |
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| author | Junesang Choi |
| author_facet | Junesang Choi |
| author_sort | Junesang Choi |
| collection | DOAJ |
| description | Ever since Euler first evaluated ζ(2) and ζ(2m), numerous interesting solutions of the problem of evaluating the ζ(2m) (m∈ℕ) have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation of ζ(2m) (m∈ℕ) is known for ζ(2m+1) (m∈ℕ) or even for any special case such as ζ(3). Instead, various rapidly converging series for ζ(2m+1) have been developed by many authors. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for ζ(2m+1). In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations of β(2m+1) and ζ(2m) (m∈ℕ), which have been treated in earlier works. |
| format | Article |
| id | doaj-art-91fb82a5624b44fb92817cc4867aec21 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-91fb82a5624b44fb92817cc4867aec212025-02-03T01:11:06ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/457620457620Rapidly Converging Series for ζ(2n+1) from Fourier SeriesJunesang Choi0Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of KoreaEver since Euler first evaluated ζ(2) and ζ(2m), numerous interesting solutions of the problem of evaluating the ζ(2m) (m∈ℕ) have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation of ζ(2m) (m∈ℕ) is known for ζ(2m+1) (m∈ℕ) or even for any special case such as ζ(3). Instead, various rapidly converging series for ζ(2m+1) have been developed by many authors. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for ζ(2m+1). In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations of β(2m+1) and ζ(2m) (m∈ℕ), which have been treated in earlier works.http://dx.doi.org/10.1155/2014/457620 |
| spellingShingle | Junesang Choi Rapidly Converging Series for ζ(2n+1) from Fourier Series Abstract and Applied Analysis |
| title | Rapidly Converging Series for ζ(2n+1) from Fourier Series |
| title_full | Rapidly Converging Series for ζ(2n+1) from Fourier Series |
| title_fullStr | Rapidly Converging Series for ζ(2n+1) from Fourier Series |
| title_full_unstemmed | Rapidly Converging Series for ζ(2n+1) from Fourier Series |
| title_short | Rapidly Converging Series for ζ(2n+1) from Fourier Series |
| title_sort | rapidly converging series for ζ 2n 1 from fourier series |
| url | http://dx.doi.org/10.1155/2014/457620 |
| work_keys_str_mv | AT junesangchoi rapidlyconvergingseriesforz2n1fromfourierseries |