# Abel criterion

From Encyclopedia of Mathematics

Abel's criterion for series of numbers. If the series

is convergent and if the numbers form a monotone bounded sequence, then the series

is convergent.

Abel's criterion for series of functions. The series

converges uniformly on a set if the series

converges uniformly on and if the functions , for any , form a monotone sequence that is uniformly bounded on . An Abel criterion for the uniform convergence of integrals

which depend on a parameter , can be formulated in a similar manner.

The Abel criteria can be strengthened (see, for example, Dedekind criterion (convergence of series)). See also Dirichlet criterion (convergence of series); Abel transformation.

#### References

[1] | G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964) |

[2] | L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian) |

[3] | E.T. Whittaker, G.N. Watson, "A course of modern analysis" , 1–2 , Cambridge Univ. Press (1952) |

**How to Cite This Entry:**

Abel criterion.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Abel_criterion&oldid=13122

This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article