View source for Weierstrass criterion (for uniform convergence)

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

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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

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such that

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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

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converges uniformly and absolutely on the entire real axis, since

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and the series

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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

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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

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

====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>



====Comments====


====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>