Weierstrass criterion (for uniform convergence)

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)

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