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A few of them are given below. For the first harmonic, the wavelength of the wave pattern would be two times the length of the string (see table above); thus, the wavelength is 160 cm or 1.60 m.The speed of the standing wave can now be determined from the wavelength and the frequency. Amer. of the Harmonic Progression as an Integral. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music. One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest power of two: Each term of the harmonic series is greater than or equal to the corresponding term of the second series, and therefore the sum of the harmonic series must be greater than or equal to the sum of the second series. 8-9, 2004. J. Thomas J. Osler, “Partial sums of series that cannot be an integer”, Riemann series theorem § Changing the sum, On-Line Encyclopedia of Integer Sequences, https://www.jstor.org/stable/24496876?seq=1#page_scan_tab_contents, "The Harmonic Series Diverges Again and Again", "Proof Without Words: The Alternating Harmonic Series Sums to ln 2", 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series), 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials), 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes), Hypergeometric function of a matrix argument, https://en.wikipedia.org/w/index.php?title=Harmonic_series_(mathematics)&oldid=987891341, Articles with specifically marked weasel-worded phrases from September 2018, Articles with unsourced statements from February 2018, Creative Commons Attribution-ShareAlike License, This page was last edited on 9 November 2020, at 21:15. And so the fuel required increases exponentially with the desired distance. The divergence, however, is very slow. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. Each rectangle is 1 unit wide and 1/n units high, so the total area of the infinite number of rectangles is the sum of the harmonic series: Additionally, the total area under the curve y = 1/x from 1 to infinity is given by a divergent improper integral: Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well. The series sum_(k=1)^infty1/k (1) is called the harmonic series. to be a high point of medieval mathematics. ... (OEIS A004080; DeTemple and Wang 1991). The value of the sum for p = 3 is called Apéry's constant, since Roger Apéry proved that it is an irrational number. the latter proof published and popularized by his brother Jacob Bernoulli. 15092688622113788323693563264538101449859497, , Boca Raton, FL: CRC Press, p. 8, The distance that can be traversed with a given amount of fuel is related to the partial sums of the harmonic series, which grow logarithmically. New York: Another problem involving the harmonic series is the Jeep problem, which (in one form) asks how much total fuel is required for a jeep with a limited fuel-carrying capacity to cross a desert, possibly leaving fuel drops along the route. Math. CRC Standard Mathematical Tables, 28th ed. The fact of this convergence is an easy consequence of either the Kolmogorov three-series theorem or of the closely related Kolmogorov maximal inequality. function. Knowledge-based programming for everyone. New York: Hyperion, p. 217, 1998. "The Harmonic Series." More precisely, the comparison above proves that. Sums of the Harmonic Series, The Sum Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Specifically, consider the arrangement of rectangles shown in the figure to the right. The Penguin Dictionary of Curious and Interesting Numbers. and since an infinite sum of 1/2's diverges, so does the harmonic series. Sum of Harmonic Sequence. Proofs were given in the 17th century by Pietro Mengoli[2] and by Johann Bernoulli,[3] Amer., pp. Arithmetic Series Calculator,Geometric Series Calculator,Harmonic Series Calculator. The depleted harmonic series where all of the terms in which the digit 9 appears anywhere in the denominator are removed can be shown to converge to the value 22.92067661926415034816....[12] In fact, when all the terms containing any particular string of digits (in any base) are removed, the series converges. Rather surprisingly, the alternating series, converges to the natural logarithm of 2. Experimentation in Mathematics: Computational Paths to Discovery. Determine Sequence Expand Sequence-- Enter Series-- (Optional) Number of Expansion terms Havil, J. taken over all primes also diverges Related 10 in Mathematical

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