1 − 2 + 3 − 4 + ⋯: Perbedaan antara revisi
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* {{cite journal |last=Ferraro |first=Giovanni |title=The First Modern Definition of the Sum of a Divergent Series: An Aspect of the Rise of 20th Century Mathematics |journal=Archive for History of Exact Sciences |year=1999 |month=June |volume=54 |issue=2 |pages=101–135 |doi=10.1007/s004070050036}} |
* {{cite journal |last=Ferraro |first=Giovanni |title=The First Modern Definition of the Sum of a Divergent Series: An Aspect of the Rise of 20th Century Mathematics |journal=Archive for History of Exact Sciences |year=1999 |month=June |volume=54 |issue=2 |pages=101–135 |doi=10.1007/s004070050036}} |
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* {{cite book|last=Grattan-Guinness|authorlink=Ivor Grattan-Guinness|first=Ivor|year=1970|title=The development of the foundations of mathematical analysis from Euler to Riemann|publisher=MIT Press|isbn= 0-262-07034-0}} |
* {{cite book|last=Grattan-Guinness|authorlink=Ivor Grattan-Guinness|first=Ivor|year=1970|title=The development of the foundations of mathematical analysis from Euler to Riemann|publisher=MIT Press|isbn= 0-262-07034-0}} |
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*{{cite book|last=Hardy|first=G. H.|authorlink=G. H. Hardy|title=Divergent Series|year=1949|publisher=Clarendon Press|lccn=49005496|oclc=808787|mr=0030620|pages=xvi+396|isbn=978-0-8218-2649-2|nopp=true}} 2nd Ed. published by Chelsea Pub. Co., 1991. {{lccn|91075377}}. ISBN |
* {{cite book|last=Hardy|first=G. H.|authorlink=G. H. Hardy|title=Divergent Series|year=1949|publisher=Clarendon Press|lccn=49005496|oclc=808787|mr=0030620|pages=xvi+396|isbn=978-0-8218-2649-2|nopp=true}} 2nd Ed. published by Chelsea Pub. Co., 1991. {{lccn|91075377}}. ISBN 0-8284-0334-1. |
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* {{cite journal |doi=10.2307/2690371 |last=Kline |first=Morris |authorlink=Morris Kline |title=Euler and Infinite Series |journal=Mathematics Magazine |volume=56 |issue=5 |year=1983 |month=November |pages=307–314 |jstor=2690371}} |
* {{cite journal |doi=10.2307/2690371 |last=Kline |first=Morris |authorlink=Morris Kline |title=Euler and Infinite Series |journal=Mathematics Magazine |volume=56 |issue=5 |year=1983 |month=November |pages=307–314 |jstor=2690371}} |
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* {{cite book|first=Shaughan|last=Lavine|title=Understanding the Infinite|year=1994|publisher=Harvard UP|isbn= 0-674-92096-1}} |
* {{cite book|first=Shaughan|last=Lavine|title=Understanding the Infinite|year=1994|publisher=Harvard UP|isbn= 0-674-92096-1}} |
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*{{cite book|last=Markusevič|first=Aleksej Ivanovič|title=Series: fundamental concepts with historical exposition|year=1967|edition=English translation of 3rd revised edition (1961) in Russian|publisher=Hindustan Pub. Corp.|lccn=sa68017528|pages=176|oclc=729238507|location =Delhi, India}} Author also known as A. I. Markushevich and Alekseï Ivanovitch Markouchevitch. Also published in Boston, Mass by Heath with {{oclc|474456247}}. Additionally, {{oclc|208730}}, {{oclc|487226828}}. |
* {{cite book|last=Markusevič|first=Aleksej Ivanovič|title=Series: fundamental concepts with historical exposition|year=1967|edition=English translation of 3rd revised edition (1961) in Russian|publisher=Hindustan Pub. Corp.|lccn=sa68017528|pages=176|oclc=729238507|location =Delhi, India}} Author also known as A. I. Markushevich and Alekseï Ivanovitch Markouchevitch. Also published in Boston, Mass by Heath with {{oclc|474456247}}. Additionally, {{oclc|208730}}, {{oclc|487226828}}. |
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* {{cite book|author=Saichev, A.I., and W.A. Woyczyński|title=Distributions in the physical and engineering sciences, Volume 1|publisher=Birkhaüser|year=1996|isbn= 0-8176-3924-1}} |
* {{cite book|author=Saichev, A.I., and W.A. Woyczyński|title=Distributions in the physical and engineering sciences, Volume 1|publisher=Birkhaüser|year=1996|isbn= 0-8176-3924-1}} |
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* {{cite journal |last=Tucciarone |first=John |title=The development of the theory of summable divergent series from 1880 to 1925 |journal=Archive for History of Exact Sciences |volume=10 |issue=1–2 |year=1973 |month=January |pages=1–40 |doi=10.1007/BF00343405}} |
* {{cite journal |last=Tucciarone |first=John |title=The development of the theory of summable divergent series from 1880 to 1925 |journal=Archive for History of Exact Sciences |volume=10 |issue=1–2 |year=1973 |month=January |pages=1–40 |doi=10.1007/BF00343405}} |
Revisi per 19 Juni 2016 07.35
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Pm1234_Ground.png/250px-Pm1234_Ground.png)
Dalam matematika, 1 − 2 + 3 − 4 + ··· adalah deret tak hingga yang suku-sukunya berupa bilangan bulat positif berurutan makin besar serta bernilai positif dan negatif secara selang-seling. Dengan notasi jumlah sigma, jumlah suku pertama m dapat dijabarkan menjadi
Deret tak hingga bersifat menyempit (divergen), artinya barisan jumlah parsialnya, (1, −1, 2, −2, ...), cenderung tidak punya batas terhingga apapun.[1][2] Namun pada abad ke-18, Leonhard Euler menulis sesuatu yang ia akui sebagai suatu persamaan paradoks:[3]
Penjelasan yang lebih teliti mengenai persamaan ini baru muncul kemudian. Sejak 1890, Ernesto Cesàro, Émile Borel, dan ilmuwan lainnya mencari metode yang terdefinisikan dengan jelas untuk menerapkan penjumlahan umum pada deret divergen—termasuk penafsiran baru mengenai metode-metode Euler.[4][5] Banyak metode keterjumlahan (summability) yang dengan mudahnya menerapkan "jumlah" ¼ pada 1 − 2 + 3 − 4 + .... Penjumlahan Cesàro adalah satu dari sedikit sekali metode yang tidak menjumlahkan 1 − 2 + 3 − 4 + ..., dan deret tersebut menjadi contoh perlunya suatu metode yang agak lebih kuat seperti penjumlahan Abel.
Deret 1 − 2 + 3 − 4 + ... sangat terkait dengan deret Grandi 1 − 1 + 1 − 1 + .... Euler menyebut keduanya sebagai kasus istimewa 1 − 2n + 3n − 4n + ... untuk n arbitrer, yaitu rangkaian penelitian yang memperluas hasil penelitiannya tentang masalah Basel dan mengarah pada persamaan fungsi yang kita kenal sebagai fungsi eta Dirichlet dan the fungsi zeta Riemann.[6]
Referensi
Catatan kaki
Daftar pustaka
- Beals, Richard (2004). Analysis: an introduction. Cambridge UP. ISBN 0-521-60047-2.
- Davis, Harry F. (1989). Fourier Series and Orthogonal Functions. Dover. ISBN 0-486-65973-9.
- Euler, Leonhard; Lucas Willis; and Thomas J Osler (2006). "Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series". The Euler Archive. Diakses tanggal 2007-03-22. Originally published as Euler, Leonhard (1768). "Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques". Memoires de l'academie des sciences de Berlin. 17: 83–106.
- Ferraro, Giovanni (1999). "The First Modern Definition of the Sum of a Divergent Series: An Aspect of the Rise of 20th Century Mathematics". Archive for History of Exact Sciences. 54 (2): 101–135. doi:10.1007/s004070050036.
- Grattan-Guinness, Ivor (1970). The development of the foundations of mathematical analysis from Euler to Riemann. MIT Press. ISBN 0-262-07034-0.
- Hardy, G. H. (1949). Divergent Series. Clarendon Press. xvi+396. ISBN 978-0-8218-2649-2. LCCN 49005496. MR 0030620. OCLC 808787. 2nd Ed. published by Chelsea Pub. Co., 1991. LCCN 91-75377. ISBN 0-8284-0334-1.
- Kline, Morris (1983). "Euler and Infinite Series". Mathematics Magazine. 56 (5): 307–314. doi:10.2307/2690371. JSTOR 2690371.
- Lavine, Shaughan (1994). Understanding the Infinite. Harvard UP. ISBN 0-674-92096-1.
- Markusevič, Aleksej Ivanovič (1967). Series: fundamental concepts with historical exposition (edisi ke-English translation of 3rd revised edition (1961) in Russian). Delhi, India: Hindustan Pub. Corp. hlm. 176. LCCN sa68017528. OCLC 729238507. Author also known as A. I. Markushevich and Alekseï Ivanovitch Markouchevitch. Also published in Boston, Mass by Heath with OCLC 474456247. Additionally, OCLC 208730, OCLC 487226828.
- Saichev, A.I., and W.A. Woyczyński (1996). Distributions in the physical and engineering sciences, Volume 1. Birkhaüser. ISBN 0-8176-3924-1.
- Tucciarone, John (1973). "The development of the theory of summable divergent series from 1880 to 1925". Archive for History of Exact Sciences. 10 (1–2): 1–40. doi:10.1007/BF00343405.
- Vretblad, Anders (2003). Fourier Analysis and Its Applications. Springer. ISBN 0-387-00836-5.
- Weidlich, John E. (1950). Summability methods for divergent series. Stanford M.S. theses. OCLC 38624384.