Abelian limit theorem
The Abelian limit theorem is a mathematical theorem from the branch of analysis . It describes the conditions under which a function defined as a power series can be continuously extended to the edges of the convergence interval and reads as follows:
- Let be a convergent series of real numbers. Then the power series converges on the interval and the function defined by it is continuous on with .
application
The inverse function of the tangent function has the following representation as a power series on the interval :
- .
The series converges according to the Leibniz criterion . There , the Abelian limit theorem provides the identity
- .
literature
- Kurt Endl, Wolfgang Luh: Analysis II. An integrated representation. 7th edition. Aula-Verlag Wiesbaden 1989, p. 205.
- Harro Heuser : Textbook of Analysis . Part 1. 6th edition. Teubner 1989, ISBN 3-519-42221-2 , p. 367.
- Vieweg Mathematics Lexicon . Vieweg-Verlag, (1988).
Web links
- Eric W. Weisstein : generalized version of Abel's limit theorem . In: MathWorld (English).
- Abel's limit theorem . In: PlanetMath . (English)
- Proof of Abel's limit theorem . In: PlanetMath . (English)