Editing Harmonic series (mathematics) (section)

==History==
[[File:Harmonic series to 32.svg|thumb|A wave and its harmonics, with wavelengths <math>1,\tfrac12,\tfrac13,\dots</math>]] 
The name of the harmonic series derives from the concept of [[overtone]]s or harmonics [[harmonic series (music)|in music]]: the [[wavelength]]s of the overtones of a vibrating string are {{nowrap|<math>\tfrac12</math>,}} {{nowrap|<math>\tfrac13</math>,}} {{nowrap|<math>\tfrac14</math>,}} etc., of the string's [[Fundamental frequency|fundamental wavelength]].{{r|rice|kullman}}  Every term of the harmonic series after the first is the [[harmonic mean]] of the neighboring terms, so the terms form a [[Harmonic progression (mathematics)|harmonic progression]]; the phrases ''harmonic mean'' and ''harmonic progression'' likewise derive from music.{{r|kullman}}
Beyond music, harmonic sequences have also had a certain popularity with architects. This was so particularly in the [[Baroque]] period, when architects used them to establish the [[Proportion (architecture)|proportions]] of [[Architectural drawing#Floor plan|floor plans]], of [[Architectural drawing#Elevation|elevations]], and to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.{{r|hersey}}

The divergence of the harmonic series was first proven in 1350 by [[Nicole Oresme]].{{r|kullman|oresme}} Oresme's work, and the contemporaneous work of [[Richard Swineshead]] on a different series, marked the first appearance of infinite series other than the [[geometric series]] in mathematics.{{r|stillwell}} However, this achievement fell into obscurity.{{r|derbyshire}} Additional proofs were published in the 17th century by [[Pietro Mengoli]]{{r|kullman|mengoli}} and by [[Jacob Bernoulli]].{{r|jacob1|jacob2|dunham}} Bernoulli credited his brother [[Johann Bernoulli]] for finding the proof,{{r|dunham}} and it was later included in Johann Bernoulli's collected works.{{r|johann}}

The partial sums of the harmonic series were named [[harmonic number]]s, and given their usual notation <math>H_n</math>, in 1968 by [[Donald Knuth]].{{r|knuth}}