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v t e

In mathematics, the harmonic series is the divergent infinite series:

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots }$

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.

Contents

 [hide] 

• 1 History
• 2 Paradoxes
• 3 Divergence
  • 3.1 Comparison test
  • 3.2 Integral test
• 4 Rate of divergence
• 5 Partial sums
• 6 Related series
  • 6.1 Alternating harmonic series
  • 6.2 General harmonic series
  • 6.3 p-series
  • 6.4 ln-series
  • 6.5 φ-series
  • 6.6 Random harmonic series
  • 6.7 Depleted harmonic series
• 7 See also
• 8 References
• 9 External links

History[edit]

The fact that the harmonic series diverges was first proven in the 14th century by Nicole Oresme,^[1] but this achievement fell into obscurity. Proofs were given in the 17th century by Pietro Mengoli,^[2] Johann Bernoulli,^[3] and Jacob Bernoulli.^[4][5]
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.^[6]

Paradoxes[edit]

The harmonic series can be counterintuitive to students first encountering it, because it is a divergent series even though the limit of the nth term as n goes to infinity is zero. The divergence of the harmonic series is also the source of some apparent paradoxes. One example of these is the "worm on the rubber band".^[7] Suppose that a worm crawls along an infinitely-elastic one-meter rubber band at the same time as the rubber band is uniformly stretched. If the worm travels 1 centimeter per minute and the band stretches 1 meter per minute, will the worm ever reach the end of the rubber band? The answer, counterintuitively, is "yes", for after n minutes, the ratio of the distance travelled by the worm to the total length of the rubber band is

${\displaystyle {\frac {1}{100}}\sum _{k=1}^{n}{\frac {1}{k}}}$