இசைத் தொடர் (கணிதம்) (Tamil Wikipedia)

Analysis of information sources in references of the Wikipedia article "இசைத் தொடர் (கணிதம்)" in Tamil language version.

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ams.org (Global: 451^st place; Tamil: 1,535^th place)

mathscinet.ams.org

Stillwell, John (2010). Mathematics and its History. Undergraduate Texts in Mathematics (3rd ed.). New York: Springer. p. 182. doi:10.1007/978-1-4419-6053-5. ISBN 978-1-4419-6052-8. MR 2667826.
Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Washington, DC: Joseph Henry Press. p. 10. ISBN 0-309-08549-7. MR 1968857.
Bressoud, David M. (2007). A Radical Approach to Real Analysis. Classroom Resource Materials Series (2nd ed.). Washington, DC: Mathematical Association of America. pp. 137–138. ISBN 978-0-88385-747-2. MR 2284828.

archive.org (Global: 6^th place; Tamil: 8^th place)

Kullman, David E. (May 2001). "What's harmonic about the harmonic series?". The College Mathematics Journal 32 (3): 201–203. doi:10.2307/2687471. https://archive.org/details/sim_college-mathematics-journal_2001-05_32_3/page/201.
Hersey, George L. (2001). Architecture and Geometry in the Age of the Baroque. University of Chicago Press. pp. 11–12, 37–51. ISBN 978-0-226-32783-9.
Stillwell, John (2010). Mathematics and its History. Undergraduate Texts in Mathematics (3rd ed.). New York: Springer. p. 182. doi:10.1007/978-1-4419-6053-5. ISBN 978-1-4419-6052-8. MR 2667826.
Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Washington, DC: Joseph Henry Press. p. 10. ISBN 0-309-08549-7. MR 1968857.
Dunham, William (January 1987). "The Bernoullis and the harmonic series". The College Mathematics Journal 18 (1): 18–23. doi:10.1080/07468342.1987.11973001. https://archive.org/details/sim_college-mathematics-journal_1987-01_18_1/page/n21.
Ralph P. Boas Jr.; John Wrench (1971). "Partial sums of the harmonic series". The American Mathematical Monthly 78 (8): 864–870. doi:10.1080/00029890.1971.11992881. https://archive.org/details/sim_american-mathematical-monthly_1971-10_78_8/page/864.

books.google.com (Global: 3^rd place; Tamil: 6^th place)

Bernoulli, Jacob (1713). Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis [Theory of inference, posthumous work. With the Treatise on infinite series…]. Basel: Thurneysen. pp. 250–251.
From p. 250, prop. 16:
"XVI. Summa serei infinita harmonicè progressionalium, ${\tfrac {1}{1}}+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{5}}$ &c. est infinita. Id primus deprehendit Frater:…"

[16. The sum of an infinite series of harmonic progression, ${\tfrac {1}{1}}+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{5}}+\cdots$ , is infinite. My brother first discovered this…]
Bernoulli, Johann (1742). "Corollary III of De seriebus varia". Opera Omnia. Lausanne & Basel: Marc-Michel Bousquet & Co. vol. 4, p. 8. Johann Bernoulli's proof is also by contradiction. It uses a telescopic sum to represent each term ${\tfrac {1}{n}}$ as ${\frac {1}{n}}={\Big (}{\frac {1}{n}}-{\frac {1}{n+1}}{\Big )}+{\Big (}{\frac {1}{n+1}}-{\frac {1}{n+2}}{\Big )}+{\Big (}{\frac {1}{n+2}}-{\frac {1}{n+3}}{\Big )}\cdots$ $={\frac {1}{n(n+1)}}+{\frac {1}{(n+1)(n+2)}}+{\frac {1}{(n+2)(n+3)}}\cdots$ Changing the order of summation in the corresponding double series gives, in modern notation $S=\sum _{n=1}^{\infty }{\frac {1}{n}}=\sum _{n=1}^{\infty }\sum _{k=n}^{\infty }{\frac {1}{k(k+1)}}=\sum _{k=1}^{\infty }\sum _{n=1}^{k}{\frac {1}{k(k+1)}}$ $=\sum _{k=1}^{\infty }{\frac {k}{k(k+1)}}=\sum _{k=1}^{\infty }{\frac {1}{k+1}}=S-1$ .
Bressoud, David M. (2007). A Radical Approach to Real Analysis. Classroom Resource Materials Series (2nd ed.). Washington, DC: Mathematical Association of America. pp. 137–138. ISBN 978-0-88385-747-2. MR 2284828.
Havil, Julian (2003). "Chapter 2: The harmonic series". Gamma: Exploring Euler's Constant. Princeton University Press. pp. 21–25. ISBN 978-0-691-14133-6.