math lessons - Harmonic series (mathematics)

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

In mathematics, the harmonic series is the infinite series

$\sum_{k=1}^\infty \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$

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. This can be proved by noting that the harmonic series is term-by-term larger than or equal to the series

$\sum_{k=1}^\infty 2^{-\lceil \log_2 k \rceil} \! = 1 + \left[\frac{1}{2}\right] + \left[\frac{1}{4} + \frac{1}{4}\right] + \left[\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right] + \frac{1}{16}\cdots$

$= \quad\ 1 +\ \frac{1}{2}\ +\ \quad\frac{1}{2} \ \quad+ \ \qquad\quad\frac{1}{2}\qquad\ \quad \ + \ \quad\ \cdots$

which clearly diverges. 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:

$\sum_{k = 1}^\infty \frac{(-1)^{k + 1}}{k} = \ln 2.$

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

If we define the n-th harmonic number as

$H_n = \sum_{k = 1}^n \frac{1}{k}$

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

$\int_1^n {1 \over x}\, dx$

whose value is ln(n).

More precisely, we have the limit:

$\lim_{n \to \infty} H_n - \ln(n) = \gamma$

where γ is the Euler-Mascheroni constant.

It can be proved 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

$\sigma(n)\le H_n + \ln(H_n)e^{H_n} \qquad \mbox{ for every }n\in\mathbb{N}$

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 generalised harmonic series, or p-series, is (any of) the series

$\sum_{n=1}^{\infty}\frac{1}{n^p}$

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.

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