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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|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w0974301.png" /></td> </tr></table>
+
$$
 +
\sum _ { n= } 1 ^  \infty 
 +
u _ {n} ( x)
 +
$$
  
of real- or complex-valued functions defined on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w0974302.png" /> there exists a convergent series of non-negative numbers
+
of real- or complex-valued functions defined on some set $  E $
 +
there exists a convergent series of non-negative numbers
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w0974303.png" /></td> </tr></table>
+
$$
 +
\sum _ { n= } 1 ^  \infty  a _ {n}  $$
  
 
such that
 
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w0974304.png" /></td> </tr></table>
+
$$
 +
| u _ {n} ( x) |  \leq  a _ {n} ,\ \
 +
n = 1, 2 \dots
 +
$$
 +
 
 +
then the initial series converges uniformly and absolutely on the set  $  E $(
 +
cf. [[Absolutely convergent series|Absolutely convergent series]]). For instance, the series
  
then the initial series converges uniformly and absolutely on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w0974305.png" /> (cf. [[Absolutely convergent series|Absolutely convergent series]]). For instance, the series
+
$$
 +
\sum _ { n= } 1 ^  \infty 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w0974306.png" /></td> </tr></table>
+
\frac{\sin  nx }{n  ^ {2} }
 +
 
 +
$$
  
 
converges uniformly and absolutely on the entire real axis, since
 
converges uniformly and absolutely on the entire real axis, since
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w0974307.png" /></td> </tr></table>
+
$$
 +
\left |
 +
\frac{\sin  nx }{n  ^ {2} }
 +
\right |
 +
\leq 
 +
\frac{1}{n  ^ {2} }
 +
 
 +
$$
  
 
and the series
 
and the series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w0974308.png" /></td> </tr></table>
+
$$
 +
\sum _ { n= } 1 ^  \infty 
 +
\frac{1}{n  ^ {2} }
 +
 
 +
$$
  
 
is convergent.
 
is convergent.
  
If, for a sequence of real- or complex-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w0974309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743010.png" /> which converges to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743011.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743012.png" /> there exists a sequence of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743015.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743016.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743019.png" /> then the sequence converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743020.png" />. For instance, the sequence
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743021.png" /></td> </tr></table>
+
$$
 +
f _ {n} ( x)  = 1 -
  
converges uniformly to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743022.png" /> on the entire real axis, since
+
\frac{(- 1)  ^ {n} }{x  ^ {2} + n }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097430/w09743023.png" /></td> </tr></table>
+
$$
 +
 
 +
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.
 
The Weierstrass criterion for uniform convergence may also be applied to functions with values in normed linear spaces.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  K. Weierstrass,  "Abhandlungen aus der Funktionenlehre" , Springer  (1866)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  K. Weierstrass,  "Math. Werke" , '''1–7''' , G. Olms &amp; Johnson, reprint  (1927)</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  K. Weierstrass,  "Abhandlungen aus der Funktionenlehre" , Springer  (1866)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  K. Weierstrass,  "Math. Werke" , '''1–7''' , G. Olms &amp; Johnson, reprint  (1927)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Knopp,  "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Knopp,  "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1974)  pp. 24</TD></TR></table>

Latest revision as of 08:28, 6 June 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