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 ${\displaystyle \sum _{n=0}^{\infty }a_{n}}$ ${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}}$ ${\displaystyle [0,1]}$ ${\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}}$ ${\displaystyle [0,1]}$ ${\displaystyle f(1)=\sum _{n=0}^{\infty }a_{n}}$ .

application

The inverse function of the tangent function has the following representation as a power series on the interval : ${\displaystyle (0,1)\subset (-1,1)}$

${\displaystyle \arctan(x)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n+1}}{2n+1}}}$.

The series converges according to the Leibniz criterion . There , the Abelian limit theorem provides the identity ${\displaystyle \textstyle \sum _{n=0}^{\infty }(-1)^{n}{\frac {1}{2n+1}}}$ ${\displaystyle \tan({\tfrac {\pi }{4}})=1}$

${\displaystyle {\frac {\pi }{4}}=\arctan(1)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {1}{2n+1}}}$.

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)