Harmonic series (mathematics) (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Harmonic series (mathematics)" in English language version.

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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.
Boas, R. P. Jr.; Wrench, J. W. Jr. (1971). "Partial sums of the harmonic series". The American Mathematical Monthly. 78 (8): 864–870. doi:10.1080/00029890.1971.11992881. JSTOR 2316476. MR 0289994.
Sanna, Carlo (2016). "On the $p$ -adic valuation of harmonic numbers". Journal of Number Theory. 166: 41–46. doi:10.1016/j.jnt.2016.02.020. hdl:2318/1622121. MR 3486261.
Ross, Bertram (1978). "The psi function". Mathematics Magazine. 51 (3): 176–179. doi:10.1080/0025570X.1978.11976704. JSTOR 2689999. MR 1572267.
Paterson, Mike; Peres, Yuval; Thorup, Mikkel; Winkler, Peter; Zwick, Uri (2009). "Maximum overhang". The American Mathematical Monthly. 116 (9): 763–787. doi:10.4169/000298909X474855. MR 2572086. S2CID 1713091.
Rubinstein-Salzedo, Simon (2017). "Could Euler have conjectured the prime number theorem?". Mathematics Magazine. 90 (5): 355–359. arXiv:1701.04718. doi:10.4169/math.mag.90.5.355. JSTOR 10.4169/math.mag.90.5.355. MR 3738242. S2CID 119165483.
Pollack, Paul (2015). "Euler and the partial sums of the prime harmonic series". Elemente der Mathematik. 70 (1): 13–20. doi:10.4171/EM/268. MR 3300350.
Tsang, Kai-Man (2010). "Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function". Science China. 53 (9): 2561–2572. Bibcode:2010ScChA..53.2561T. doi:10.1007/s11425-010-4068-6. hdl:10722/129254. MR 2718848. S2CID 6168120.
Frieze, Alan; Karoński, Michał (2016). "4.1 Connectivity". Introduction to Random Graphs. Cambridge University Press, Cambridge. pp. 64–68. doi:10.1017/CBO9781316339831. ISBN 978-1-107-11850-8. MR 3675279.
Isaac, Richard (1995). "8.4 The coupon collector's problem solved". The Pleasures of Probability. Undergraduate Texts in Mathematics. New York: Springer-Verlag. pp. 80–82. doi:10.1007/978-1-4612-0819-8. ISBN 0-387-94415-X. MR 1329545.
Freniche, Francisco J. (2010). "On Riemann's rearrangement theorem for the alternating harmonic series" (PDF). The American Mathematical Monthly. 117 (5): 442–448. doi:10.4169/000298910X485969. JSTOR 10.4169/000298910x485969. MR 2663251. S2CID 20575373.
Soddy, F. (1943). "The three infinite harmonic series and their sums (with topical reference to the Newton and Leibniz series for $\pi$ )". Proceedings of the Royal Society. 182 (989): 113–129. Bibcode:1943RSPSA.182..113S. doi:10.1098/rspa.1943.0026. MR 0009207. S2CID 202575422.
Bombieri, E. (2010). "The classical theory of zeta and $L$ -functions". Milan Journal of Mathematics. 78 (1): 11–59. doi:10.1007/s00032-010-0121-8. MR 2684771. S2CID 120058240.
Bettin, Sandro; Molteni, Giuseppe; Sanna, Carlo (2018). "Small values of signed harmonic sums". Comptes Rendus Mathématique. 356 (11–12): 1062–1074. arXiv:1806.05402. doi:10.1016/j.crma.2018.11.007. hdl:2434/634047. MR 3907571. S2CID 119160796.

arxiv.org

Rubinstein-Salzedo, Simon (2017). "Could Euler have conjectured the prime number theorem?". Mathematics Magazine. 90 (5): 355–359. arXiv:1701.04718. doi:10.4169/math.mag.90.5.355. JSTOR 10.4169/math.mag.90.5.355. MR 3738242. S2CID 119165483.
Bettin, Sandro; Molteni, Giuseppe; Sanna, Carlo (2018). "Small values of signed harmonic sums". Comptes Rendus Mathématique. 356 (11–12): 1062–1074. arXiv:1806.05402. doi:10.1016/j.crma.2018.11.007. hdl:2434/634047. MR 3907571. S2CID 119160796.

books.google.com

Mengoli, Pietro (1650). "Praefatio [Preface]". Novae quadraturae arithmeticae, seu De additione fractionum [New arithmetic quadrature (i.e., integration), or On the addition of fractions] (in Latin). Bologna: Giacomo Monti. Mengoli's proof is by contradiction: Let $S$ denote the sum of the series. Group the terms of the series in triplets: $S=1+({\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}})+({\tfrac {1}{5}}+{\tfrac {1}{6}}+{\tfrac {1}{7}})+\cdots$ . Since for $x>1$ , ${\tfrac {1}{x-1}}+{\tfrac {1}{x}}+{\tfrac {1}{x+1}}>{\tfrac {3}{x}}$ , then $S>1+{\tfrac {3}{3}}+{\tfrac {3}{6}}+{\tfrac {3}{9}}+\cdots =1+1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\cdots =1+S$ , which is impossible for any finite $S$ . Therefore, the series diverges.
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.

doi.org

Kullman, David E. (May 2001). "What's harmonic about the harmonic series?". The College Mathematics Journal. 32 (3): 201–203. doi:10.2307/2687471. JSTOR 2687471.
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.
Dunham, William (January 1987). "The Bernoullis and the harmonic series". The College Mathematics Journal. 18 (1): 18–23. doi:10.1080/07468342.1987.11973001. JSTOR 2686312.
Boas, R. P. Jr.; Wrench, J. W. Jr. (1971). "Partial sums of the harmonic series". The American Mathematical Monthly. 78 (8): 864–870. doi:10.1080/00029890.1971.11992881. JSTOR 2316476. MR 0289994.
Osler, Thomas J. (November 2012). "96.53 Partial sums of series that cannot be an integer". The Mathematical Gazette. 96 (537): 515–519. doi:10.1017/S0025557200005167. JSTOR 24496876. S2CID 124359670. See in particular Theorem 1, p. 516.
Sanna, Carlo (2016). "On the $p$ -adic valuation of harmonic numbers". Journal of Number Theory. 166: 41–46. doi:10.1016/j.jnt.2016.02.020. hdl:2318/1622121. MR 3486261.
Ross, Bertram (1978). "The psi function". Mathematics Magazine. 51 (3): 176–179. doi:10.1080/0025570X.1978.11976704. JSTOR 2689999. MR 1572267.
Sofo, Anthony; Srivastava, H. M. (2015). "A family of shifted harmonic sums". The Ramanujan Journal. 37: 89–108. doi:10.1007/s11139-014-9600-9. S2CID 254990799.
Hadley, John; Singmaster, David (March 1992). "Problems to sharpen the young: An annotated translation of Propositiones ad acuendos juvenes". The Mathematical Gazette. 76 (475): 102–126. doi:10.2307/3620384. JSTOR 3620384. S2CID 125835186. See problem 52: De homine patrefamilias – A lord of the manor, pp. 124–125.
Gale, David (May 1970). "The jeep once more or jeeper by the dozen". The American Mathematical Monthly. 77 (5): 493–501. doi:10.1080/00029890.1970.11992525. JSTOR 2317382.
Paterson, Mike; Peres, Yuval; Thorup, Mikkel; Winkler, Peter; Zwick, Uri (2009). "Maximum overhang". The American Mathematical Monthly. 116 (9): 763–787. doi:10.4169/000298909X474855. MR 2572086. S2CID 1713091.
Rubinstein-Salzedo, Simon (2017). "Could Euler have conjectured the prime number theorem?". Mathematics Magazine. 90 (5): 355–359. arXiv:1701.04718. doi:10.4169/math.mag.90.5.355. JSTOR 10.4169/math.mag.90.5.355. MR 3738242. S2CID 119165483.
Pollack, Paul (2015). "Euler and the partial sums of the prime harmonic series". Elemente der Mathematik. 70 (1): 13–20. doi:10.4171/EM/268. MR 3300350.
Tsang, Kai-Man (2010). "Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function". Science China. 53 (9): 2561–2572. Bibcode:2010ScChA..53.2561T. doi:10.1007/s11425-010-4068-6. hdl:10722/129254. MR 2718848. S2CID 6168120.
Maunsell, F. G. (October 1938). "A problem in cartophily". The Mathematical Gazette. 22 (251): 328–331. doi:10.2307/3607889. JSTOR 3607889. S2CID 126381029.
Gerke, Oke (April 2013). "How much is it going to cost me to complete a collection of football trading cards?". Teaching Statistics. 35 (2): 89–93. doi:10.1111/test.12005. S2CID 119887116.
Luko, Stephen N. (March 2009). "The "coupon collector's problem" and quality control". Quality Engineering. 21 (2): 168–181. doi:10.1080/08982110802642555. S2CID 109194745.
Frieze, Alan; Karoński, Michał (2016). "4.1 Connectivity". Introduction to Random Graphs. Cambridge University Press, Cambridge. pp. 64–68. doi:10.1017/CBO9781316339831. ISBN 978-1-107-11850-8. MR 3675279.
Isaac, Richard (1995). "8.4 The coupon collector's problem solved". The Pleasures of Probability. Undergraduate Texts in Mathematics. New York: Springer-Verlag. pp. 80–82. doi:10.1007/978-1-4612-0819-8. ISBN 0-387-94415-X. MR 1329545.
Freniche, Francisco J. (2010). "On Riemann's rearrangement theorem for the alternating harmonic series" (PDF). The American Mathematical Monthly. 117 (5): 442–448. doi:10.4169/000298910X485969. JSTOR 10.4169/000298910x485969. MR 2663251. S2CID 20575373.
Soddy, F. (1943). "The three infinite harmonic series and their sums (with topical reference to the Newton and Leibniz series for $\pi$ )". Proceedings of the Royal Society. 182 (989): 113–129. Bibcode:1943RSPSA.182..113S. doi:10.1098/rspa.1943.0026. MR 0009207. S2CID 202575422.
Bombieri, E. (2010). "The classical theory of zeta and $L$ -functions". Milan Journal of Mathematics. 78 (1): 11–59. doi:10.1007/s00032-010-0121-8. MR 2684771. S2CID 120058240.
Schmuland, Byron (May 2003). "Random harmonic series" (PDF). The American Mathematical Monthly. 110 (5): 407–416. doi:10.2307/3647827. JSTOR 3647827. Archived from the original (PDF) on 2011-06-08. Retrieved 2006-08-07.
Bettin, Sandro; Molteni, Giuseppe; Sanna, Carlo (2018). "Small values of signed harmonic sums". Comptes Rendus Mathématique. 356 (11–12): 1062–1074. arXiv:1806.05402. doi:10.1016/j.crma.2018.11.007. hdl:2434/634047. MR 3907571. S2CID 119160796.
Baillie, Robert (May 1979). "Sums of reciprocals of integers missing a given digit". The American Mathematical Monthly. 86 (5): 372–374. doi:10.1080/00029890.1979.11994810. JSTOR 2321096.
Schmelzer, Thomas; Baillie, Robert (June 2008). "Summing a curious, slowly convergent series". The American Mathematical Monthly. 115 (6): 525–540. doi:10.1080/00029890.2008.11920559. JSTOR 27642532. S2CID 11461182.

gla.ac.uk

eprints.gla.ac.uk

Freniche, Francisco J. (2010). "On Riemann's rearrangement theorem for the alternating harmonic series" (PDF). The American Mathematical Monthly. 117 (5): 442–448. doi:10.4169/000298910X485969. JSTOR 10.4169/000298910x485969. MR 2663251. S2CID 20575373.

handle.net

hdl.handle.net

Sanna, Carlo (2016). "On the $p$ -adic valuation of harmonic numbers". Journal of Number Theory. 166: 41–46. doi:10.1016/j.jnt.2016.02.020. hdl:2318/1622121. MR 3486261.
Tsang, Kai-Man (2010). "Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function". Science China. 53 (9): 2561–2572. Bibcode:2010ScChA..53.2561T. doi:10.1007/s11425-010-4068-6. hdl:10722/129254. MR 2718848. S2CID 6168120.
Bettin, Sandro; Molteni, Giuseppe; Sanna, Carlo (2018). "Small values of signed harmonic sums". Comptes Rendus Mathématique. 356 (11–12): 1062–1074. arXiv:1806.05402. doi:10.1016/j.crma.2018.11.007. hdl:2434/634047. MR 3907571. S2CID 119160796.

harvard.edu

ui.adsabs.harvard.edu

Tsang, Kai-Man (2010). "Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function". Science China. 53 (9): 2561–2572. Bibcode:2010ScChA..53.2561T. doi:10.1007/s11425-010-4068-6. hdl:10722/129254. MR 2718848. S2CID 6168120.
Soddy, F. (1943). "The three infinite harmonic series and their sums (with topical reference to the Newton and Leibniz series for $\pi$ )". Proceedings of the Royal Society. 182 (989): 113–129. Bibcode:1943RSPSA.182..113S. doi:10.1098/rspa.1943.0026. MR 0009207. S2CID 202575422.

jstor.org

Kullman, David E. (May 2001). "What's harmonic about the harmonic series?". The College Mathematics Journal. 32 (3): 201–203. doi:10.2307/2687471. JSTOR 2687471.
Dunham, William (January 1987). "The Bernoullis and the harmonic series". The College Mathematics Journal. 18 (1): 18–23. doi:10.1080/07468342.1987.11973001. JSTOR 2686312.
Roy, Ranjan (December 2007). "Review of A Radical Approach to Real Analysis by David M. Bressoud". SIAM Review. 49 (4): 717–719. JSTOR 20454048. One might point out that Cauchy's condensation test is merely the extension of Oresme's argument for the divergence of the harmonic series
Boas, R. P. Jr.; Wrench, J. W. Jr. (1971). "Partial sums of the harmonic series". The American Mathematical Monthly. 78 (8): 864–870. doi:10.1080/00029890.1971.11992881. JSTOR 2316476. MR 0289994.
Osler, Thomas J. (November 2012). "96.53 Partial sums of series that cannot be an integer". The Mathematical Gazette. 96 (537): 515–519. doi:10.1017/S0025557200005167. JSTOR 24496876. S2CID 124359670. See in particular Theorem 1, p. 516.
Ross, Bertram (1978). "The psi function". Mathematics Magazine. 51 (3): 176–179. doi:10.1080/0025570X.1978.11976704. JSTOR 2689999. MR 1572267.
Hadley, John; Singmaster, David (March 1992). "Problems to sharpen the young: An annotated translation of Propositiones ad acuendos juvenes". The Mathematical Gazette. 76 (475): 102–126. doi:10.2307/3620384. JSTOR 3620384. S2CID 125835186. See problem 52: De homine patrefamilias – A lord of the manor, pp. 124–125.
Gale, David (May 1970). "The jeep once more or jeeper by the dozen". The American Mathematical Monthly. 77 (5): 493–501. doi:10.1080/00029890.1970.11992525. JSTOR 2317382.
Rubinstein-Salzedo, Simon (2017). "Could Euler have conjectured the prime number theorem?". Mathematics Magazine. 90 (5): 355–359. arXiv:1701.04718. doi:10.4169/math.mag.90.5.355. JSTOR 10.4169/math.mag.90.5.355. MR 3738242. S2CID 119165483.
Maunsell, F. G. (October 1938). "A problem in cartophily". The Mathematical Gazette. 22 (251): 328–331. doi:10.2307/3607889. JSTOR 3607889. S2CID 126381029.
Freniche, Francisco J. (2010). "On Riemann's rearrangement theorem for the alternating harmonic series" (PDF). The American Mathematical Monthly. 117 (5): 442–448. doi:10.4169/000298910X485969. JSTOR 10.4169/000298910x485969. MR 2663251. S2CID 20575373.
Schmuland, Byron (May 2003). "Random harmonic series" (PDF). The American Mathematical Monthly. 110 (5): 407–416. doi:10.2307/3647827. JSTOR 3647827. Archived from the original (PDF) on 2011-06-08. Retrieved 2006-08-07.
Baillie, Robert (May 1979). "Sums of reciprocals of integers missing a given digit". The American Mathematical Monthly. 86 (5): 372–374. doi:10.1080/00029890.1979.11994810. JSTOR 2321096.
Schmelzer, Thomas; Baillie, Robert (June 2008). "Summing a curious, slowly convergent series". The American Mathematical Monthly. 115 (6): 525–540. doi:10.1080/00029890.2008.11920559. JSTOR 27642532. S2CID 11461182.

pacific.edu

scholarlycommons.pacific.edu

Euler, Leonhard (1737). "Variae observationes circa series infinitas" [Various observations concerning infinite series]. Commentarii Academiae Scientiarum Petropolitanae (in Latin). 9: 160–188.

pme-math.org

Sharp, R. T. (1954). "Problem 52: Overhanging dominoes" (PDF). Pi Mu Epsilon Journal. 1 (10): 411–412.

semanticscholar.org

api.semanticscholar.org

Osler, Thomas J. (November 2012). "96.53 Partial sums of series that cannot be an integer". The Mathematical Gazette. 96 (537): 515–519. doi:10.1017/S0025557200005167. JSTOR 24496876. S2CID 124359670. See in particular Theorem 1, p. 516.
Sofo, Anthony; Srivastava, H. M. (2015). "A family of shifted harmonic sums". The Ramanujan Journal. 37: 89–108. doi:10.1007/s11139-014-9600-9. S2CID 254990799.
Hadley, John; Singmaster, David (March 1992). "Problems to sharpen the young: An annotated translation of Propositiones ad acuendos juvenes". The Mathematical Gazette. 76 (475): 102–126. doi:10.2307/3620384. JSTOR 3620384. S2CID 125835186. See problem 52: De homine patrefamilias – A lord of the manor, pp. 124–125.
Paterson, Mike; Peres, Yuval; Thorup, Mikkel; Winkler, Peter; Zwick, Uri (2009). "Maximum overhang". The American Mathematical Monthly. 116 (9): 763–787. doi:10.4169/000298909X474855. MR 2572086. S2CID 1713091.
Rubinstein-Salzedo, Simon (2017). "Could Euler have conjectured the prime number theorem?". Mathematics Magazine. 90 (5): 355–359. arXiv:1701.04718. doi:10.4169/math.mag.90.5.355. JSTOR 10.4169/math.mag.90.5.355. MR 3738242. S2CID 119165483.
Tsang, Kai-Man (2010). "Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function". Science China. 53 (9): 2561–2572. Bibcode:2010ScChA..53.2561T. doi:10.1007/s11425-010-4068-6. hdl:10722/129254. MR 2718848. S2CID 6168120.
Maunsell, F. G. (October 1938). "A problem in cartophily". The Mathematical Gazette. 22 (251): 328–331. doi:10.2307/3607889. JSTOR 3607889. S2CID 126381029.
Gerke, Oke (April 2013). "How much is it going to cost me to complete a collection of football trading cards?". Teaching Statistics. 35 (2): 89–93. doi:10.1111/test.12005. S2CID 119887116.
Luko, Stephen N. (March 2009). "The "coupon collector's problem" and quality control". Quality Engineering. 21 (2): 168–181. doi:10.1080/08982110802642555. S2CID 109194745.
Freniche, Francisco J. (2010). "On Riemann's rearrangement theorem for the alternating harmonic series" (PDF). The American Mathematical Monthly. 117 (5): 442–448. doi:10.4169/000298910X485969. JSTOR 10.4169/000298910x485969. MR 2663251. S2CID 20575373.
Soddy, F. (1943). "The three infinite harmonic series and their sums (with topical reference to the Newton and Leibniz series for $\pi$ )". Proceedings of the Royal Society. 182 (989): 113–129. Bibcode:1943RSPSA.182..113S. doi:10.1098/rspa.1943.0026. MR 0009207. S2CID 202575422.
Bombieri, E. (2010). "The classical theory of zeta and $L$ -functions". Milan Journal of Mathematics. 78 (1): 11–59. doi:10.1007/s00032-010-0121-8. MR 2684771. S2CID 120058240.
Bettin, Sandro; Molteni, Giuseppe; Sanna, Carlo (2018). "Small values of signed harmonic sums". Comptes Rendus Mathématique. 356 (11–12): 1062–1074. arXiv:1806.05402. doi:10.1016/j.crma.2018.11.007. hdl:2434/634047. MR 3907571. S2CID 119160796.
Schmelzer, Thomas; Baillie, Robert (June 2008). "Summing a curious, slowly convergent series". The American Mathematical Monthly. 115 (6): 525–540. doi:10.1080/00029890.2008.11920559. JSTOR 27642532. S2CID 11461182.

stevekifowit.com

Kifowit, Steven J.; Stamps, Terra A. (Spring 2006). "The harmonic series diverges again and again" (PDF). AMATYC Review. 27 (2). American Mathematical Association of Two-Year Colleges: 31–43. See also unpublished addendum, "More proofs of divergence of the harmonic series" by Kifowit.

ualberta.ca

stat.ualberta.ca

Schmuland, Byron (May 2003). "Random harmonic series" (PDF). The American Mathematical Monthly. 110 (5): 407–416. doi:10.2307/3647827. JSTOR 3647827. Archived from the original (PDF) on 2011-06-08. Retrieved 2006-08-07.

web.archive.org

Schmuland, Byron (May 2003). "Random harmonic series" (PDF). The American Mathematical Monthly. 110 (5): 407–416. doi:10.2307/3647827. JSTOR 3647827. Archived from the original (PDF) on 2011-06-08. Retrieved 2006-08-07.

youtube.com

Parker, Matt (February 12, 2022). "The coupon collector's problem (with Geoff Marshall)". Stand-up maths. YouTube.