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Harmonic series (mathematics)
In mathematics, the harmonic series is the 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 term harmonic mean likewise derives from music.
[ ] Divergence of the harmonic series
The harmonic series diverges to infinity, albeit rather slowly (the first 1043 terms sum to less than 100 [1]). One way to prove this divergence is by noting that the harmonic series is term-by-term larger than or equal to another divergent series:

The sum of infinitely many 1⁄2 terms clearly diverges to infinity and therefore the harmonic series also diverges. More precisely, if is the 2k-th partial sum of the harmonic series, then

which clearly diverges, although slowly (at a logarithmic rate). This proof, due to Nicole Oresme, is a high point of medieval mathematics. It is still a standard proof taught in mathematics classes today. Cauchy's condensation test is a generalization of this argument.
Another proof uses the integral test for convergence, relating the harmonic series to the (divergent) integral of 1⁄x over the interval from 1 to infinity.
Even the sum of the reciprocals of just the prime numbers diverges to infinity, although at an exponentially slower rate; known proofs of this fact are much more difficult.
[ ] Alternate proof of divergence
Suppose that the Harmonic series converges to a sum, S:

Then, redistributing the fractions, leaves

Simplifying the second group yields

Substituting the second group for S leaves

This then comes to the conclusion that

or the conclusion

This certainly cannot...