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In mathematics, the harmonic series is the divergent infinite series: : Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase ''harmonic mean'' likewise derives from music. ==History== The fact that the harmonic series diverges was first proven in the 14th century by Nicole Oresme,〔Nicole Oresme (ca. 1360) ''Quastiones super Geometriam Euclidis'' (Questions concerning Euclid's Geometry).〕 but this achievement fell into obscurity. Proofs were given in the 17th century by Pietro Mengoli,〔Pietro Mengoli, ''Novæ quadraturæ arithmeticæ, seu De additione fractionum'' (arithmetic quadrature (i.e., integration), or On the addition of fractions ) (Bologna ("Bononiæ"), (Italy): Giacomo Monti ("Jacobi Monti"), 1650). The proof of the divergence of the harmonic series is presented in the book's (preface (Præfatio) ). Mengoli's proof is by contradiction: Let S denote the sum of the series. Group the terms of the series in triplets: S = 1 + (1/2 + 1/3 + 1/4) + (1/5 + 1/6 + 1/7) + (1/8 + 1/9 + 1/10) + … Since for x > 1, 1/(x-1) + 1/x + 1/(x+1) > 3/x, then S > 1 + (3/3) + (3/6) + (3/9) + … = 1 + 1 + 1/2 + 1/3 + … = 1 + S, which is false for any finite S. Therefore, the series diverges.〕 Johann Bernoulli,〔See: Corollary III of ''De seriebus varia'' in: Johannis Bernoulli, ''Opera Omnia'' (Lausanne & Basel, Switzerland: Marc-Michel Bousquet & Co., 1742), vol. 4, (p. 8, Corollary III. )〕 and Jacob Bernoulli.〔See: * Jacob Bernoulli, ''Propositiones arithmeticae de seriebus infinitis earumque summa finita'' (propositions about infinite series and their finite sums ) (Basel, Switzerland: J. Conrad, 1689). * Jacob Bernoulli, ''Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis'' … (of inference, posthumous work. With the "Treatise on infinite series" joined … ) (Basel, Switzerland: Thurneysen, 1713), (pp. 250–251. ) From page 250, proposition 16: ''"XVI. Summa serei infinita harmonicè progressionalium, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 &c. est infinita.'' :''Id primus deprehendit Frater: … "'' (16. The sum of an infinite series of harmonic progression, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + … , is infinite. () brother first discovered it (this proof ).)〕 Historically, harmonic sequences have had a certain popularity with architects. This was so particularly in the Baroque period, when architects used them to establish the proportions of floor plans, of elevations, and to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.〔George L. Hersey, ''Architecture and Geometry in the Age of the Baroque'', p 11-12 and p37-51.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Harmonic series (mathematics)」の詳細全文を読む スポンサード リンク
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