Kummer criterion
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
A general convergence criterion for series with positive terms, proposed by E. Kummer. Given a series
 | (*) |
and an arbitrary sequence 
of positive numbers such that the series 
is divergent. If there exists an 
such that for 
,
 |
where 
is a constant positive number, then the series (*) is convergent. If 
for 
, the series (*) is divergent.
In terms of limits Kummer's criterion may be stated as follows. Let
 |
then the series (*) is convergent if 
and divergent if 
.
References
| [1] | G.M. Fichtenholz, "Differential und Integralrechnung" , 2 , Deutsch. Verlag Wissenschaft. (1964) |
Comments
References
| [a1] | E.D. Rainville, "Infinite series" , Macmillan (1967) |
How to Cite This Entry:
Kummer criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kummer_criterion&oldid=14698
Kummer criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kummer_criterion&oldid=14698
This article was adapted from an original article by E.G. Sobolevskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article