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Difference between revisions of "Weierstrass criterion (for uniform convergence)"

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A theorem which gives sufficient conditions for the [[Uniform convergence|uniform convergence]] of a [[Series|series]] or sequence of functions by comparing them with appropriate series and sequences of numbers; established by K. Weierstrass . If, for the series
+
A theorem which gives sufficient conditions for the [[uniform convergence]] of a [[series]] or sequence of functions by comparing them with appropriate series and sequences of numbers; established by K. Weierstrass . If, for the series
  
 
$$  
 
$$  
\sum _ { n= } 1 ^  \infty   
+
\sum _ { n= 1} ^  \infty  u _ {n} ( x)
u _ {n} ( x)
 
 
$$
 
$$
  
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$$  
 
$$  
\sum _ { n= } 1 ^  \infty  a _ {n}  $$
+
\sum _ { n= 1} ^  \infty  a _ {n}  $$
  
 
such that
 
such that
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$$
 
$$
  
then the initial series converges uniformly and absolutely on the set  $  E $(
+
then the initial series converges uniformly and absolutely on the set  $  E $
cf. [[Absolutely convergent series|Absolutely convergent series]]). For instance, the series
+
(cf. [[Absolutely convergent series]]). For instance, the series
  
 
$$  
 
$$  
\sum _ { n= } 1 ^  \infty   
+
\sum _ { n= 1} ^  \infty   
  
 
\frac{\sin  nx }{n  ^ {2} }
 
\frac{\sin  nx }{n  ^ {2} }
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$$  
 
$$  
\sum _ { n= } 1 ^  \infty   
+
\sum _ { n= 1} ^  \infty  \frac{1}{n  ^ {2} }
\frac{1}{n  ^ {2} }
 
 
 
 
$$
 
$$
 
 
is convergent.
 
is convergent.
  
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$$  
 
$$  
f _ {n} ( x)  =  1 -
+
f _ {n} ( x)  =  1 - \frac{(- 1)  ^ {n} }{x  ^ {2} + n }
 
 
\frac{(- 1)  ^ {n} }{x  ^ {2} + n }
 
  
 
$$
 
$$

Latest revision as of 20:50, 19 December 2020


A theorem which gives sufficient conditions for the uniform convergence of a series or sequence of functions by comparing them with appropriate series and sequences of numbers; established by K. Weierstrass . If, for the series

$$ \sum _ { n= 1} ^ \infty u _ {n} ( x) $$

of real- or complex-valued functions defined on some set $ E $ there exists a convergent series of non-negative numbers

$$ \sum _ { n= 1} ^ \infty a _ {n} $$

such that

$$ | u _ {n} ( x) | \leq a _ {n} ,\ \ n = 1, 2 \dots $$

then the initial series converges uniformly and absolutely on the set $ E $ (cf. Absolutely convergent series). For instance, the series

$$ \sum _ { n= 1} ^ \infty \frac{\sin nx }{n ^ {2} } $$

converges uniformly and absolutely on the entire real axis, since

$$ \left | \frac{\sin nx }{n ^ {2} } \right | \leq \frac{1}{n ^ {2} } $$

and the series

$$ \sum _ { n= 1} ^ \infty \frac{1}{n ^ {2} } $$ is convergent.

If, for a sequence of real- or complex-valued functions $ {f _ {n} } $, $ n = 1, 2 \dots $ which converges to a function $ f $ on a set $ E $ there exists a sequence of numbers $ \alpha _ {n} $, $ \alpha _ {n} > 0 $ and $ \alpha _ {n} \rightarrow 0 $ as $ n \rightarrow \infty $, such that $ | f( x) - {f _ {n} } ( x) | \leq \alpha _ {n} $, $ x \in E $, $ n = 1, 2 \dots $ then the sequence converges uniformly on $ E $. For instance, the sequence

$$ f _ {n} ( x) = 1 - \frac{(- 1) ^ {n} }{x ^ {2} + n } $$

converges uniformly to the function $ f( x) = 1 $ on the entire real axis, since

$$ | 1- f _ {n} ( x) | < \frac{1}{n} \ \textrm{ and } \ \lim\limits _ {n \rightarrow \infty } \ \frac{1}{n} = 0 . $$

The Weierstrass criterion for uniform convergence may also be applied to functions with values in normed linear spaces.

References

[1a] K. Weierstrass, "Abhandlungen aus der Funktionenlehre" , Springer (1866)
[1b] K. Weierstrass, "Math. Werke" , 1–7 , G. Olms & Johnson, reprint (1927)

Comments

References

[a1] T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974)
[a2] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)
[a3] W. Rudin, "Real and complex analysis" , McGraw-Hill (1974) pp. 24
How to Cite This Entry:
Weierstrass criterion (for uniform convergence). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_criterion_(for_uniform_convergence)&oldid=49190
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article