See harmonic series (music) for the (related) musical concept.
In mathematics, the harmonic series is the infinite series
- <math>\sum_{k=1}^\infty \frac{1}{k} =
1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} +
\cdots </math>
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
- <math>\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 </math>
- <math> = \quad\ 1 +\ \frac{1}{2}\ +\ \quad\frac{1}{2} \ \quad+ \ \qquad\quad\frac{1}{2}\qquad\ \quad \ + \ \quad\ \cdots </math>
which clearly diverges. Even the sum of the reciprocals of the prime numbers diverges to infinity (although that is much harder to prove; see here).
The alternating harmonic series converges however:
- <math>\sum_{k = 1}^\infty \frac{(-1)^{k + 1}}{k} = \ln 2.</math>
This is a consequence of the
Taylor series of the
natural logarithm.
If we define the <math>n</math>th harmonic number as
- <math>H_n = \sum_{k = 1}^n \frac{1}{k}</math>
then
Hn grows about as fast as the
natural logarithm of <math>n</math>. The reason is that the sum is approximated by the
integral
- <math>\int_1^n {1 \over x}\, dx</math>
whose value is ln(
n).
More precisely, we have the limit:
- <math> \lim_{n \to \infty} H_n - \ln(n) = \gamma</math>
where γ is the
Euler-Mascheroni constant.
Lagarias proved in 2001 that the Riemann hypothesis is equivalent to the statement
- <math>\sigma(n)\le H_n + \ln(H_n)e^{H_n} \qquad \mbox{ for every }n\in\mathbb{N}</math>
where σ(
n) stands for the sum of positive
divisors of
n.