# 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).