Harmonic series (mathematics)

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See harmonic series (music) for the (related) musical concept.

In mathematics, the harmonic series is the infinite series

[\sum_^\infty \frac = 1 + \frac + \frac + \frac +\cdots. ]

It is so called because the wavelengths of the overtones of a vibrating string are proportional to 1, 1/2, 1/3, 1/4, ... .

It diverges, albeit slowly, to infinity. One way to prove this is by noting that the harmonic series is term-by-term larger than or equal to the series

[\sum_^\infty 2^ \! = 1 + \left[fracright] + \left[frac + fracright] + \left[frac + frac + frac + fracright] + \frac\cdots ]

::[ = \quad\ 1 +\ \frac\ +\ \quad\frac \ \quad+ \ \qquad\quad\frac\qquad\ \quad \ + \ \quad\ \cdots ]

which clearly diverges. (This proof, due to Nicole Oresme, is a high point of medieval mathematics.) Even the sum of the reciprocals of the prime numbers diverges to infinity (although that is much harder to prove; see proof that the sum of the reciprocals of the primes diverges). The alternating harmonic series converges however:

[\sum_^\infty \frac} = \ln 2.]

This is a consequence of the Taylor series of the natural logarithm.

If we define the n-th harmonic number as

[H_n = \sum_^n \frac]

then H_n grows about as fast as the natural logarithm of n. The reason is that the sum is approximated by the integral

[\int_1^n \, dx]

whose value is ln(n).

More precisely, we have the limit:

[ \lim_ H_n - \ln(n) = \gamma]

where γ is the Euler-Mascheroni constant.

It has been proven that:

The only H_n that is an integer is H₁.
The difference H_m − H_n where m > n is never an integer.

Jeffrey Lagarias proved in 2001 that the Riemann hypothesis is equivalent to the statement

[\sigma(n)\le H_n + \ln(H_n)e^ \qquad \mboxn\in\mathbb]

where σ(n) stands for the sum of positive divisors of n. (See An Elementary Problem Equivalent to the Riemann Hypothesis, American Mathematical Monthly, volume 109 (2002), pages 534--543.)

The general harmonic series is of the form

[\sum_^\frac ]

All general harmonic series diverge.

The p-series, is (any of) the series

[\sum_^\frac ]

for p a positive real number. The series is convergent if p > 1 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.

This can be used in the testing of convergence of series.

Harmonic series (mathematics)

See also