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:''See harmonic series (music) for the (related) musical concept.''
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 (see harmonic series (music)). Every term of the series after the first is the harmonic mean of the neighboring terms; the term "harmonic mean" also is derived ultimately from music.

Contents

Divergence of the harmonic series

Convergence of the alternating harmonic series

Partial sums

General harmonic series

"''p''-series"

Random harmonic series

Depleted harmonic series

See also

Notes

Divergence of the harmonic series

The harmonic series diverges, albeit rather slowly, to infinity (the first 10⁴³ terms sum to less than 100). One way to prove the divergence is by noting that the harmonic series is term-by-term larger than or equal to another divergent series
:
1 + left[rac{1}{2}
ight] + left[rac{1}{3} + rac{1}{4}
ight] + left[rac{1}{5} + rac{1}{6} + rac{1}{7} + rac{1}{8}
ight] + left[rac{1}{9}+cdots
ight.
:::$quad\; ge\; sum\_\{k=1\}^infty\; 2^\{-lceil\; log\_2\; k$
ceil},!
:::
ight] + left[rac{1}{4} + rac{1}{4}
ight]
+ left[rac{1}{8} + rac{1}{8} + rac{1}{8} + rac{1}{8}
ight] + left[rac{1}{16}+cdots
ight.,!
:::
which clearly diverges. (Both sets of grouping can rigorously be imposed since all terms in each series have the same sign.) This proof, due to Nicole Oresme, is a high point of medieval mathematics.
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 are much more difficult; see proof that the sum of the reciprocals of the primes diverges for details.

Convergence of the alternating harmonic series

The 'alternating harmonic series' converges:
: $$
sum_{k = 1}^infty rac{(-1)^{k + 1}}{k} = 1 - rac{1}{2} + rac{1}{3} - rac{1}{4} + cdots
= ln 2 = 0.69314718dots

This equality is a consequence of the Mercator series, the Taylor series for the natural logarithm.
Another very interesting equality, similar in form to Mercator's series is:
:$$
sum_{k=0}^infty rac{(-1)^k}{2k+1} = 1 - rac{1}{3} + rac{1}{5} -rac{1}{7} +cdots = rac{pi}{4}.

Partial sums

The ''n''th partial sum of the diverging harmonic series,
:
is called the ''n''th 'harmonic number'.
The difference between distinct harmonic numbers is never an integer.
Jeffrey Lagarias proved in 2001 that the Riemann hypothesis is logically equivalent to the statement
:$sigma(n)le\; H\_n\; +\; ln(H\_n)e^\{H\_n\}\; qquad\; ext\{\; for\; every\; \}ninmathbb\{N\}$
where σ(''n'') stands for the sum of the positive divisors of ''n''.^[1]

General harmonic series

The 'general harmonic series' is of the form
:
All general harmonic series diverge.

"''p''-series"

The '''p''-series' is (any of) the series
:
for ''p'' a positive real number. The series is always convergent if ''p'' > 1 (when it is called the 'over-harmonic series') and divergent otherwise. When ''p'' = 1, the series is the harmonic series. If ''p'' > 1 then the sum of the series is ζ(''p''), i.e., the Riemann zeta function evaluated at ''p''.

Random harmonic series

Byron Schmuland of the University of Alberta examined^[2][3] the properties of the random harmonic series
:
where the ''s''_''n'' are independent, identically distributed random variables taking the values +1 and −1 with equal probability 1/2. He shows that this sum converges with probability 1 and that the convergent is a random variable with some interesting properties. In particular, the probability density function of this random variable evaluated at +2 or at −2 takes on the value 0.1249999999999999999999999999999999999999997642..., differing from 1/8 by less than 10⁻⁴². Schmuland's paper actually explains why this probability is so close to, but not exactly, 1/8.

Depleted harmonic series

The depleted harmonic series where all of the terms with a 9 in the denominator are removed actually can be shown to converge and its value is less than 80.^[4][5]

See also

★ Complex logarithm

★ Harmonic mean

★ Harmonic number

★ Riemann zeta function

Notes

1. ''An Elementary Problem Equivalent to the Riemann Hypothesis'', American Mathematical Monthly, volume 109 (2002), pages 534--543.
2. "Random Harmonic Series", ''American Mathematical Monthly'' 110, 407-416, May 2003
3. Schmuland's preprint of ''Random Harmonic Series''
4. http://www.qbyte.org/puzzles/p072a.html
5. http://www.qbyte.org/puzzles/p072s.html