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Uniform Convergence

A sequence of functions {f_n}, n=1, 2, 3, ... is said to be uniformly convergent to f for a set E of values of x if, for each epsilon>0, an integer N can be found such that

|f_n(x)-f(x)|<epsilon
(1)

for n>=N and all x in E.

A series sumf_n(x) converges uniformly on E if the sequence {S_n} of partial sums defined by

sum_(k=1)^nf_k(x)=S_n(x)
(2)

converges uniformly on E.

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

1. The series sum

f(x)=sum_(n=1)^inftyu_n(x)
(3)

is continuous.

2. The series may be integrated term by term

int_a^bf(x)dx=sum_(n=1)^inftyint_a^bu_n(x)dx.
(4)

For example, a power series sum_(n=0)^(infty)a_n(x-x_0)^n 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 sum_(n=1)^(infty)u_n(x) does not tell anything about convergence of sum_(n=1)^(infty)d/(dx)u_n(x). Suppose that sum_(n=1)^(infty)u_n(x_0) converges for some x_0 in [a,b], that each u_n(x) is differentiable on [a,b], and that sum_(n=1)^(infty)d/(dx)u_n(x) converges uniformly on [a,b]. Then sum_(n=1)^(infty)u_n(x) converges uniformly on [a,b] to a function f, and for each x in [a,b],

d/(dx)f(x)=sum_(n=1)^inftyd/(dx)u_n(x).
(5)

See also

Abel's Convergence Theorem, Abel's Uniform Convergence Test, Weierstrass M-Test

Portions of this entry contributed by John Derwent

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 299-301, 1985.Jeffreys, H. and Jeffreys, B. S. "Uniform Convergence of Sequences and Series" et seq. §1.112-1.1155 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 37-43, 1988.Knopp, K. "Uniform Convergence." §18 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I. New York: Dover, pp. 71-73, 1996.Rudin, W. Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, pp. 147-148, 1976.

Referenced on Wolfram|Alpha

Uniform Convergence

Cite this as:

Derwent, John and Weisstein, Eric W. "Uniform Convergence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UniformConvergence.html

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