The series of numbers
Each term of the harmonic series (beginning with the second) is the harmonic mean of its two contiguous terms (hence the name harmonic series). The harmonic series is divergent (G. Leibniz, 1673), and its partial sums
increase as (L. Euler, 1740). There exists a constant , known as the Euler constant, such that , where . The series
is called the generalized harmonic series; it is convergent for and divergent for .
Generalized harmonic series are often used to test whether a given series is convergent or divergent by estimating in terms of the order of the terms of the given series; see Series.
|[a1]||G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979)|
Harmonic series. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Harmonic_series&oldid=17912