harmonic series
The harmonic series is
The harmonic series is known to diverge. This can be proven via the integral test; compare with
The harmonic series is a special case of the -series, , which has the form
where is some positive real number. The series is known to converge (leading to the p-series test for series convergence) iff . In using the comparison test, one can often compare a given series with positive terms to some .
Remark 1. One could call with an overharmonic series and with an underharmonic series; the corresponding names are known at least in Finland.
Remark 2. A -series is sometimes called a harmonic series, so that the harmonic series is a harmonic series with .
For complex-valued , , the Riemann zeta function.
A famous -series is (or ), which converges to . In general no -series of odd has been solved analytically.
A -series which is not summed to , but instead is of the form
is called a -series (or a harmonic series) of order of .
Title | harmonic series |
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Canonical name | HarmonicSeries |
Date of creation | 2013-03-22 13:02:46 |
Last modified on | 2013-03-22 13:02:46 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 8 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 40A05 |
Related topic | HarmonicNumber |
Related topic | PrimeHarmonicSeries |
Related topic | SumOfPowers |
Defines | p-series |
Defines | harmonic series of order |