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In [[mathematics]], the '''Weierstrass M-test''' is a test for showing that an [[infinite series]] of [[function (mathematics)|functions]] converges [[uniform convergence|uniformly]]. It applies to series whose terms are functions with [[real number|real]] or [[complex number|complex]] values, and is analogous to the [[comparison test]] for determining the convergence of series of real or complex numbers.
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The Weierstrass M-test is a special case of [[Dominated convergence theorem|Lebesgue's dominated convergence theorem]], where the [[measure]] is taken to be the [[counting measure]] over an [[Atom (measure theory)|atomic]] measure space.
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== Statement ==
<blockquote>'''Weierstrass M-test.''' Suppose that {''f<sub>n</sub>''} is a [[sequence]] of real- or complex-valued functions defined on a [[Set (mathematics)|set]] ''A'', and that there is a sequence of positive numbers {''M<sub>n</sub>''} satisfying
:<math>\forall n \geq 1, \forall x \in A \ : \ |f_n(x)|\leq M_n,</math>
:<math>\sum_{n=1}^{\infty} M_n < \infty</math>
Then the series
:<math>\sum_{n=1}^{\infty} f_n (x)</math>
converges [[uniform convergence|uniformly]] on ''A''.</blockquote>
 
 
 
'''Remark.''' The result is often used in combination with the [[uniform limit theorem]]. Together they say that if, in addition to the above conditions, the set ''A'' is a topological space and the functions ''f<sub>n</sub>'' are continuous on ''A'', then the series converges to a continuous function.
 
== Generalization ==
A more general version of the Weierstrass M-test holds if the [[codomain]] of the functions {''f<sub>n</sub>''} is any [[Banach space]], in which case the statement
:<math>|f_n|\leq M_n</math>
may be replaced by  
:<math>\|f_n\|\leq M_n</math>,
where <math>\|\cdot\|</math> is the norm on the Banach space. For an example of the use of this test on a Banach space, see the article [[Fréchet derivative]].
 
== Proof ==
Let ''M'' be the limit of the sum <math>\sum_{n=1}^{\infty} M_n</math>. Since <math>|f_n(x)| \le M_n</math> the sum <math>\sum_{n=1}^{\infty} f_n(x)</math> is absolutely convergent, call its limit ''f''(''x'').
 
By convergence of the ''M'' sum, for ε > 0 there exists an integer ''K'' such that
:<math>\forall k > K \ : \ \left|M - \sum_{n=1}^{k} M_n\right| < \varepsilon.</math>
We will show that <math>\sum_{n=1}^{\infty} f_n(x)</math> converges uniformly by showing that
:<math>\forall k > K, \forall x \in A \ : \ \left|f(x) - \sum_{n=1}^{k} f_n(x)\right| < \varepsilon.</math>
The crucial point here is that ''K'' does not depend on ''x''.
:<math>\forall x \in A \ : \ \left|f(x) - \sum_{n=1}^{k} f_n(x)\right| = \left|\sum_{n=k+1}^{\infty} f_n(x)\right| \le \sum_{n=k+1}^{\infty} \left|f_n(x)\right| \le \sum_{n=k+1}^{\infty} M_n = \left|M - \sum_{n=1}^{k} M_n\right| < \varepsilon.</math>
 
==See also==
* [[Uniform convergence#Exponential function|Example of Weierstrass M-test]]
 
==References==
*{{cite book |last=Rudin |first=Walter |title=Functional Analysis |date=January 1991 |publisher=McGraw-Hill Science/Engineering/Math |isbn=0-07-054236-8}}
*{{cite book |last=Rudin |first=Walter |title=Real and Complex Analysis |date=May 1986 |publisher=McGraw-Hill Science/Engineering/Math |isbn=0-07-054234-1}}
*{{cite book |last=Rudin |first=Walter |title=Principles of Mathematical Analysis |year=1976 |publisher=McGraw-Hill Science/Engineering/Math}}
*[[E. T. Whittaker|Whittaker]] and [[G. N. Watson|Watson]] (1927). ''A Course in Modern Analysis'', fourth edition. Cambridge University Press, p. 49.
 
[[Category:Functional analysis]]
[[Category:Mathematical series]]
[[Category:Convergence tests]]
[[Category:Articles containing proofs]]

Latest revision as of 22:14, 27 November 2014

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