1 + 2 + 3 + 4 + ⋯ (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "1 + 2 + 3 + 4 + ⋯" in English language version.

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ams.org

Sondow, Jonathan (February 1994), "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series", Proceedings of the American Mathematical Society, 120 (4): 421–424, doi:10.1090/S0002-9939-1994-1172954-7, retrieved February 14, 2014.

archive.org

Promoting numbers to functions is identified as one of two broad classes of summation methods, including Abel and Borel summation, by Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. pp. 475–476. ISBN 0-486-66165-2.
Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. pp. 490–492. ISBN 0-486-66165-2.

arxiv.org

Lepowsky, J. (1999). "Vertex operator algebras and the zeta function". In Naihuan Jing and Kailash C. Misra (ed.). Recent Developments in Quantum Affine Algebras and Related Topics. Contemporary Mathematics. Vol. 248. pp. 327–340. arXiv:math/9909178. Bibcode:1999math......9178L..
Tong, David (February 23, 2012). "String Theory". pp. 28–48. arXiv:0908.0333 [hep-th].
Kaneko, Masanobu; Kurokawa, Nobushige; Wakayama, Masato (2003), "A variation of Euler's approach to values of the Riemann zeta function" (PDF), Kyushu Journal of Mathematics, 57 (1): 175–192, arXiv:math/0206171, doi:10.2206/kyushujm.57.175, S2CID 54514141, archived from the original (PDF) on 2014-02-02, retrieved January 31, 2014.

books.google.com

Aiyangar, Srinivasa Ramanujan (7 September 1995). Ramanujan: Letters and Commentary. p. 53. ISBN 9780821891254.
Zeidler, Eberhard (2007), "Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists", Quantum Field Theory I: Basics in Mathematics and Physics. A Bridge Between Mathematicians and Physicists, Springer: 305–306, Bibcode:2006qftb.book.....Z, ISBN 9783540347644.
Complicite (April 2012), A Disappearing Number, Oberon, ISBN 9781849432993.

dartmouth.edu

math.dartmouth.edu

Euler, Leonhard (2006). "Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series". Translated by Willis, Lucas; Osler, Thomas J. The Euler Archive. Retrieved 2007-03-22. Originally published as Euler, Leonhard (1768). "Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques". Mémoires de l'Académie des Sciences de Berlin (in French). 17: 83–106.

doi.org

Barbiellini, Bernardo (1987), "The Casimir effect in conformal field theories", Physics Letters B, 190 (1–2): 137–139, Bibcode:1987PhLB..190..137B, doi:10.1016/0370-2693(87)90854-9.
Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine, 56 (5): 307–314, CiteSeerX 10.1.1.639.6923, doi:10.2307/2690371, JSTOR 2690371.
Ayoub, Raymond (December 1974), "Euler and the Zeta Function" (PDF), The American Mathematical Monthly, 81 (10): 1067–1086, doi:10.2307/2319041, JSTOR 2319041, retrieved February 14, 2014.
Kaneko, Masanobu; Kurokawa, Nobushige; Wakayama, Masato (2003), "A variation of Euler's approach to values of the Riemann zeta function" (PDF), Kyushu Journal of Mathematics, 57 (1): 175–192, arXiv:math/0206171, doi:10.2206/kyushujm.57.175, S2CID 54514141, archived from the original (PDF) on 2014-02-02, retrieved January 31, 2014.
Sondow, Jonathan (February 1994), "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series", Proceedings of the American Mathematical Society, 120 (4): 421–424, doi:10.1090/S0002-9939-1994-1172954-7, retrieved February 14, 2014.
Barbeau, E. J.; Leah, P. J. (May 1976), "Euler's 1760 paper on divergent series", Historia Mathematica, 3 (2): 141–160, doi:10.1016/0315-0860(76)90030-6.

handle.net

hdl.handle.net

Natiello, Mario A.; Solari, Hernan Gustavo (July 2015), "On the removal of infinities from divergent series", Philosophy of Mathematics Education Journal, 29: 1–11, hdl:11336/46148.

harvard.edu

ui.adsabs.harvard.edu

Lepowsky, J. (1999). "Vertex operator algebras and the zeta function". In Naihuan Jing and Kailash C. Misra (ed.). Recent Developments in Quantum Affine Algebras and Related Topics. Contemporary Mathematics. Vol. 248. pp. 327–340. arXiv:math/9909178. Bibcode:1999math......9178L..
Barbiellini, Bernardo (1987), "The Casimir effect in conformal field theories", Physics Letters B, 190 (1–2): 137–139, Bibcode:1987PhLB..190..137B, doi:10.1016/0370-2693(87)90854-9.
Zeidler, Eberhard (2007), "Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists", Quantum Field Theory I: Basics in Mathematics and Physics. A Bridge Between Mathematicians and Physicists, Springer: 305–306, Bibcode:2006qftb.book.....Z, ISBN 9783540347644.

imsc.res.in

Ramanujan's Notebooks, retrieved January 26, 2014

jstor.org

Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine, 56 (5): 307–314, CiteSeerX 10.1.1.639.6923, doi:10.2307/2690371, JSTOR 2690371.
Ayoub, Raymond (December 1974), "Euler and the Zeta Function" (PDF), The American Mathematical Monthly, 81 (10): 1067–1086, doi:10.2307/2319041, JSTOR 2319041, retrieved February 14, 2014.

kyushu-u.ac.jp

catalog.lib.kyushu-u.ac.jp

Kaneko, Masanobu; Kurokawa, Nobushige; Wakayama, Masato (2003), "A variation of Euler's approach to values of the Riemann zeta function" (PDF), Kyushu Journal of Mathematics, 57 (1): 175–192, arXiv:math/0206171, doi:10.2206/kyushujm.57.175, S2CID 54514141, archived from the original (PDF) on 2014-02-02, retrieved January 31, 2014.

maa.org

Ayoub, Raymond (December 1974), "Euler and the Zeta Function" (PDF), The American Mathematical Monthly, 81 (10): 1067–1086, doi:10.2307/2319041, JSTOR 2319041, retrieved February 14, 2014.

maths.org

plus.maths.org

Thomas, Rachel (December 1, 2008), "A disappearing number", Plus, retrieved February 5, 2014.

nottingham.ac.uk

Padilla, Tony, What do we get if we sum all the natural numbers?, retrieved February 3, 2014.

nytimes.com

Overbye, Dennis (February 3, 2014), "In the End, It All Adds Up to –1/12", The New York Times, retrieved February 3, 2014.

psu.edu

citeseerx.ist.psu.edu

Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine, 56 (5): 307–314, CiteSeerX 10.1.1.639.6923, doi:10.2307/2690371, JSTOR 2690371.

semanticscholar.org

api.semanticscholar.org

Kaneko, Masanobu; Kurokawa, Nobushige; Wakayama, Masato (2003), "A variation of Euler's approach to values of the Riemann zeta function" (PDF), Kyushu Journal of Mathematics, 57 (1): 175–192, arXiv:math/0206171, doi:10.2206/kyushujm.57.175, S2CID 54514141, archived from the original (PDF) on 2014-02-02, retrieved January 31, 2014.

smithsonianmag.com

Schultz, Colin (2014-01-31). "The Great Debate Over Whether 1 + 2 + 3 + 4... + ∞ = −1/12". Smithsonian. Retrieved 2016-05-16.

terrytao.wordpress.com

Tao, Terence (April 10, 2010), The Euler–Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014.

u-strasbg.fr

irem.u-strasbg.fr

Lefort, Jean, "Les séries divergentes chez Euler" (PDF), L'Ouvert (in French) (31), IREM de Strasbourg: 15–25, archived from the original (PDF) on February 22, 2014, retrieved February 14, 2014.

web.archive.org

Lefort, Jean, "Les séries divergentes chez Euler" (PDF), L'Ouvert (in French) (31), IREM de Strasbourg: 15–25, archived from the original (PDF) on February 22, 2014, retrieved February 14, 2014.
Kaneko, Masanobu; Kurokawa, Nobushige; Wakayama, Masato (2003), "A variation of Euler's approach to values of the Riemann zeta function" (PDF), Kyushu Journal of Mathematics, 57 (1): 175–192, arXiv:math/0206171, doi:10.2206/kyushujm.57.175, S2CID 54514141, archived from the original (PDF) on 2014-02-02, retrieved January 31, 2014.

wikiversity.org

en.wikiversity.org

See v:Quantum mechanics/Casimir effect in one dimension.^{[unreliable source?]}

youtube.com

ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = –1/12 on YouTube.
Sum of Natural Numbers (second proof and extra footage) on YouTube.
Polster, Burkard (January 13, 2018). Numberphile v. Math: the truth about 1+2+3+...=-1/12. Retrieved August 31, 2023 – via YouTube.