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harmonic series (Definition)

The harmonic series is

$\displaystyle h = \sum_{n=1}^\infty \frac{1}{n} $

The harmonic series is known to diverge. This can be proven via the integral test; compare $ h$ with

$\displaystyle \int_{1}^\infty \frac{1}{x} \; dx. $

The harmonic series is a special case of the $ p$-series, $ h_p$, which has the form

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

where $ p$ is some positive real number. The series is known to converge (leading to the p-series test for series convergence) iff $ p > 1$. In using the comparison test, one can often compare a given series with positive terms to some $ h_p$.

Remark 1. One could call $ h_p$ with $ p > 1$ an overharmonic series and $ h_p$ with $ p < 1$ an underharmonic series; the corresponding names are known at least in Finland.

Remark 2. A $ p$-series is sometimes called a harmonic series, so that the harmonic series is a harmonic series with $ p=1$.

For complex-valued $ p$, $ h_p = \zeta(p)$, the Riemann zeta function.

A famous $ p$-series is $ h_2$ (or $ \zeta(2)$), which converges to $ \frac{\pi^2}{6}$. In general no $ p$-series of odd $ p$ has been solved analytically.

A $ p$-series which is not summed to $ \infty$, but instead is of the form

$\displaystyle h_p(k) = \sum_{n=1}^k \frac{1}{n^p} $

is called a $ p$-series (or a harmonic series) of order $ k$ of $ p$.



"harmonic series" is owned by CWoo. [ full author list (3) | owner history (1) ]
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See Also: harmonic number, prime harmonic series, sum of powers

Also defines:  p-series, harmonic series of order

Attachments:
proof of divergence of harmonic series (by grouping terms) (Proof) by rspuzio
proof of divergence of harmonic series (by splitting odd and even terms) (Definition) by rspuzio
alternating harmonic series (Example) by CWoo
a series related to harmonic series (Example) by pahio
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Cross-references: order, odd, Riemann zeta function, terms, comparison test, iff, p-series test, converge, series, real number, positive, integral test, diverge
There are 19 references to this entry.

This is version 5 of harmonic series, born on 2002-09-09, modified 2007-10-18.
Object id is 3449, canonical name is HarmonicSeries.
Accessed 10988 times total.

Classification:
AMS MSC: 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
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