In
mathematics, the Weierstrass M-test is a test for determining whether an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mat ...
of
functions converges
uniformly and
absolutely. It applies to series whose terms are
bounded function
In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that
:, f(x), \le M
for all ''x'' in ''X''. A ...
s with
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex values, and is analogous to the
comparison test for determining the convergence of series of real or complex numbers. It is named after the German mathematician
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
(1815-1897).
Statement
Weierstrass M-test.
Suppose that (''f''
''n'') is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
of real- or complex-valued functions defined on a
set ''A'', and that there is a sequence of non-negative numbers (''M''
''n'') satisfying the conditions
*
for all
and all
, and
*
converges.
Then the series
:
converges
absolutely and
uniformly on ''A''.
The result is often used in combination with the
uniform limit theorem
In mathematics, the uniform limit theorem states that the uniform limit of any sequence of continuous functions is continuous.
Statement
More precisely, let ''X'' be a topological space, let ''Y'' be a metric space, and let ƒ''n'' :&nb ...
. Together they say that if, in addition to the above conditions, the set ''A'' is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and the functions ''f
n'' are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
on ''A'', then the series converges to a continuous function.
Proof
Consider the sequence of functions
:
Since the series
converges and for every , then by the
Cauchy criterion
The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'Analy ...
,
:
For the chosen ,
:
:
(Inequality (1) follows from the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
.)
The sequence is thus a
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
in R or C, and by
completeness, it converges to some number that depends on ''x''. For ''n'' > ''N'' we can write
:
Since ''N'' does not depend on ''x'', this means that the sequence of partial sums converges uniformly to the function ''S''. Hence, by definition, the series
converges uniformly.
Analogously, one can prove that
converges uniformly.
Generalization
A more general version of the Weierstrass M-test holds if the common
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
of the functions (''f
n'') is a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vec ...
, in which case the premise
:
is to be replaced by
:
,
where
is the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
on the Banach space. For an example of the use of this test on a Banach space, see the article
Fréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued function of a single real variable to the case of a vector-valued ...
.
See also
*
Example of Weierstrass M-test
References
*
*
*
* {{cite book , last=Whittaker , first=E.T. , author-link=E. T. Whittaker , last2=Watson , first2=G.N. , author-link2=G. N. Watson , year=1927 , title=A Course in Modern Analysis , edition=Fourth , publisher=Cambridge University Press , page=49
Functional analysis
Convergence tests
Articles containing proofs