The harmonic series is
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 .
|Date of creation||2013-03-22 13:02:46|
|Last modified on||2013-03-22 13:02:46|
|Last modified by||CWoo (3771)|
|Defines||harmonic series of order|