 TOPICS # Uniform Convergence

A sequence of functions , , 2, 3, ... is said to be uniformly convergent to for a set of values of if, 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. The series may be integrated term by term (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 differentiable on , and that converges uniformly on . Then converges uniformly on to a function , and for each , (5)

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 Web Resource. https://mathworld.wolfram.com/UniformConvergence.html