quality:corruption
History

The name of the harmonic series derives from the concept of overtones or harmonics in music: the wavelengths of the overtones of a vibrating string are 1 2 {\displaystyle {\tfrac {1}{2}}}, 1 3 {\displaystyle {\tfrac {1}{3}}}, 1 4 {\displaystyle {\tfrac {1}{4}}}, etc., of the string's fundamental wavelength. Every term of the harmonic series after the first is the harmonic mean of the neighboring terms, so the terms form a harmonic progression; the phrases harmonic mean and harmonic progression likewise derive from music. 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 proportions of floor plans, of elevations, and to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.

The divergence of the harmonic series was first proven in 1350 by Nicole 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. However, this achievement fell into obscurity. Additional proofs were published in the 17th century by Pietro Mengoli and by Jacob Bernoulli. Bernoulli credited his brother Johann Bernoulli for finding the proof, and it was later included in Johann Bernoulli's collected works.

The partial sums of the harmonic series were named harmonic numbers, and given their usual notation H n {\displaystyle H_{n}}, in 1968 by Donald Knuth.