Weierstrass criterion (for uniform convergence)

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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

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

such that

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

converges uniformly and absolutely on the entire real axis, since

and the series

is convergent.

If, for a sequence of real- or complex-valued functions , which converges to a function on a set there exists a sequence of numbers , and as , such that , , then the sequence converges uniformly on . For instance, the sequence

converges uniformly to the function on the entire real axis, since

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


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



[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). L.D. Kudryavtsev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098