Weierstraß's majorant criterion

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The Weierstrasse Majorant Criterion (also: Weierstrasse M-Test ) is a criterion for the proof of uniform and absolute convergence of a series of functions . As a special case it contains the majorant criterion for rows. It was named after the mathematician Karl Weierstrasse .

statement

Let be a sequence of real or complex valued functions on the set . Let be real constants such that

applies to everyone and everyone in . Furthermore, the series converges .

Then the following applies: The series

converges absolutely and evenly on .

example

Let be a real number, then is the Weierstrass function

steady everywhere but nowhere differentiable. The continuity of this function can be demonstrated by the Weierstrass M test. It is true

such as

according to the formula for the geometric series. Therefore, the series converges uniformly according to the Weierstrass M test. The individual partial sums now form a sequence of continuous functions that converge uniformly to . Thus, as such a limit value is continuous.

literature

  • Herbert Amann and Joachim Escher, Analysis 1 , Birkhäuser, Basel, 2002. (see Theorem V.1.6)

Individual evidence

  1. H. Heuser: Textbook of Analysis Part 1 . Vieweg + Teubner (2009), sentence 105.3, p. 555.
  2. ^ EM Stein, R. Shakarchi: Fourier Analysis. An Introduction. University Press Group Ltd (2003), Theorem 3.1, p. 114.