Weierstrass M-test
Encyclopedia
In mathematics
, the Weierstrass M-test is a criterion for determining whether an infinite series of functions
converges uniformly. It applies to series whose terms are functions with real
or complex
values, and is analogous to the comparison test
for determining the convergence of series of real or complex numbers.
Suppose
is a sequence
of real- or complex-valued functions defined on a set
, and that there exist positive constants 
such that
for all
≥
and all 
, where the series
converges.
Conclusion
Then, the series
converges uniformly on
.
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
is a topological space and the functions 
are continuous on 
, then the series converges to a continuous function.
of the functions
is any Banach space
, in which case the statement
may be replaced by
,
where
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
.
converges uniformly on 
.
First note that, since
converges, the comparison test
gives that
is absolutely convergent. Therefore, 
converges pointwise on 
to some 
.
Again because
converges, for every 
there exists an 
such that
Thus, if
represents the sum of the first 
terms of 
,
Since this shows that the tail of
is less than 
independently of the value of 
, uniform convergence is proved.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Weierstrass M-test is a criterion for determining whether an infinite series of functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
converges uniformly. It applies to series whose terms are functions with real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
values, and is analogous to the comparison test
Comparison test
In mathematics, the comparison test, sometimes called the direct comparison test or CQT is a criterion for convergence or divergence of a series whose terms are real or complex numbers...
for determining the convergence of series of real or complex numbers.
Statement
ConditionsSuppose
is a sequenceSequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of real- or complex-valued functions defined on a set
, and that there exist positive constants such thatfor all
≥ and all , where the seriesconverges.
Conclusion
Then, the series
converges uniformly on
.Remark
The result is often used in combination with the uniform limit theorem
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 ƒn : X → Y be a sequence of functions converging...
. Together they say that if, in addition to the above conditions, the set
is a topological space and the functions are continuous on , then the series converges to a continuous function.Generalization
A more general version of the Weierstrass M-test holds if the codomainCodomain
In mathematics, the codomain or target set 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...
of the functions
is any Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
, in which case the statement
may be replaced by
,where
is the norm on the Banach space. For an example of the use of this test on a Banach space, see the article Fréchet derivativeFréchet derivative
In mathematics, the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from...
.
Proof
Suppose that the conditions stated above hold. We must show that converges uniformly on .First note that, since
converges, the comparison testComparison test
In mathematics, the comparison test, sometimes called the direct comparison test or CQT is a criterion for convergence or divergence of a series whose terms are real or complex numbers...
gives that
is absolutely convergent. Therefore, converges pointwise on to some .Again because
converges, for every there exists an such that-
whenever 
.
Thus, if
represents the sum of the first terms of ,-
, for all 
whenever 
Since this shows that the tail of
is less than independently of the value of , uniform convergence is proved.