Harmonic series (mathematics) (Simple English Wikipedia)

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

refsWebsite

Global rank Simple English rank

3books.google.com

3^rd place

2^nd place

26^th place

27^th place

2web.archive.org

1^st place

1^st place

742^nd place

125^th place

2^nd place

3^rd place

3,600^th place

9,556^th place

513^th place

25^th place

low place

low place

low place

low place

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]. 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: $S=1+({\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}})+({\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}})+({\frac {1}{8}}+{\frac {1}{9}}+{\frac {1}{10}})+\ldots$ Since for $x>1$ , ${\frac {1}{x-1}}+{\frac {1}{x}}+{\frac {1}{x+1}}>{\frac {3}{x}}$ , then $S>1+{\frac {3}{3}}+{\frac {3}{6}}+{\frac {3}{9}}+\ldots =1+1+{\frac {1}{2}}+{\frac {1}{3}}+\ldots =1+S$ , which is false for any finite $S$ . Therefore, the series diverges.
Bernoulli, Johann (1742). "Corollary III of De seriebus varia". Opera Omnia. Lausanne & Basel: Marc-Michel Bousquet & Co. vol. 4, p. 8.
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, ${\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}$ [et]c. est infinita. Id primus deprehendit Frater:…"

[XVI. The sum of an infinite series of harmonic progression, ${\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\ldots$ , is infinite. My brother first discovered this:…]

doi.org

Schmuland, Byron (May 2003). "Random Harmonic Series" (PDF). American Mathematical Monthly. 110 (5): 407–416. doi:10.2307/3647827. JSTOR 3647827. Archived from the original (PDF) on 2011-06-08. Retrieved 2019-12-20.

jstor.org

Thomas J. Osler, “Partial sums of series that cannot be an integer”, The Mathematical Gazette 96, November 2012, 515–519. https://www.jstor.org/stable/24496876?seq=1#page_scan_tab_contents
Schmuland, Byron (May 2003). "Random Harmonic Series" (PDF). American Mathematical Monthly. 110 (5): 407–416. doi:10.2307/3647827. JSTOR 3647827. Archived from the original (PDF) on 2011-06-08. Retrieved 2019-12-20.

oeis.org

Sloane, N. J. A. (ed.). "Sequence A082912 (Sum of a(n) terms of harmonic series is > 10ⁿ)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

pme-math.org

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

qbyte.org

"Nick's Mathematical Puzzles: Solution 72". Archived from the original on 2010-09-28. Retrieved 2019-12-20.

ualberta.ca

stat.ualberta.ca

Schmuland, Byron (May 2003). "Random Harmonic Series" (PDF). American Mathematical Monthly. 110 (5): 407–416. doi:10.2307/3647827. JSTOR 3647827. Archived from the original (PDF) on 2011-06-08. Retrieved 2019-12-20.

web.archive.org

Schmuland, Byron (May 2003). "Random Harmonic Series" (PDF). American Mathematical Monthly. 110 (5): 407–416. doi:10.2307/3647827. JSTOR 3647827. Archived from the original (PDF) on 2011-06-08. Retrieved 2019-12-20.
"Nick's Mathematical Puzzles: Solution 72". Archived from the original on 2010-09-28. Retrieved 2019-12-20.

wolfram.com

mathworld.wolfram.com

Eric W. Weisstein, Infinite Cosine Product Integral at MathWorld.