He also wrote in the Music of the Primes.............................that they had not made an equation to the West.........................................but that everything out East determines what was out West........and vice versa...................and that the zeta series............out to the East leveled out at about 1 above sea level............overlapping the pf with the neg intergers................for the 1st ten numbers........gives you ten ones...............positive ones.............................

Harmonic series (mathematics)

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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]