Uniform Convergence

Uniform convergence of a sequence of functions , , 2, 3, ... to on a set means that, for each , an integer can be found such that

(1)

for and all .

A series converges uniformly on if the sequence of partial sums defined by

(2)

converges uniformly on .

To test for uniform convergence, use Abel's uniform convergence test or the Weierstrass M-test. If individual terms of a uniformly converging series are continuous, then the following conditions are satisfied.

1. The series sum

(3)

is continuous.

2. On a finite closed interval , the series may be integrated term by term (Hunter 2014, p. 254)

(4)

For example, a power series is uniformly convergent on any closed and bounded subset inside its circle of convergence.

3. The situation is more complicated for differentiation since uniform convergence of does not tell anything about convergence of . Suppose that converges for some , that each is continuous on and differentiable on , and that converges uniformly on . Then converges uniformly on to a function , and for each (Hunter 2014, pp. 176-177),

(5)