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)