Encyclopedia > Harmonic series (mathematics)

  Article Content

Harmonic series (mathematics)

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.

See also



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
U.S. presidential election, 1804

... elections: 1792, 1796, 1800, 1804, 1808, 1812, 1816 Source: U.S. Office of the Federal R ...

 
 
 
This page was created in 28.4 ms