Abelian limit theorem

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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 .


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



  • 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).

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