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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

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