Sequence of functions [img width=22,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline1.gif[/img], [img width=31,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline2.gif[/img], 2, 3, ... is said to be uniformly convergent to [img width=9,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline3.gif[/img] for a set [img width=9,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline4.gif[/img] of values of [img width=7,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline5.gif[/img] if, for each [img width=27,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline6.gif[/img], an [url=https://mathworld.wolfram.com/Integer.html]integer[/url] [img width=11,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline7.gif[/img] can be found such that[table][tr][td][img width=98,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/NumberedEquation1.gif[/img][br][br][/td][td][br][/td][/tr][/table]for [img width=33,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline8.gif[/img] and all [img width=31,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline9.gif[/img].A [url=https://mathworld.wolfram.com/Series.html]series[/url] [img width=42,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline10.gif[/img] converges uniformly on [img width=9,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline11.gif[/img] if the sequence [img width=23,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline12.gif[/img] of partial sums defined by[table][tr][td][img width=98,height=45]https://mathworld.wolfram.com/images/equations/UniformConvergence/NumberedEquation2.gif[/img][br][br][/td][td][br][/td][/tr][/table]converges uniformly on [img width=9,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline13.gif[/img].To test for uniform convergence, use [url=https://mathworld.wolfram.com/AbelsUniformConvergenceTest.html]Abel's uniform convergence test[/url] or the [url=https://mathworld.wolfram.com/WeierstrassM-Test.html]Weierstrass M-test[/url]. If individual terms [img width=31,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline14.gif[/img] of a uniformly converging series are continuous, then the following conditions are satisfied.1. The series sum[table][tr][td][img width=95,height=44]https://mathworld.wolfram.com/images/equations/UniformConvergence/NumberedEquation3.gif[/img][br][br][/td][td][br][/td][/tr][/table]is continuous.2. The series may be integrated term by term[table][tr][td][img width=181,height=44]https://mathworld.wolfram.com/images/equations/UniformConvergence/NumberedEquation4.gif[/img][br][br][/td][td][br][/td][/tr][/table]For example, a [url=https://mathworld.wolfram.com/PowerSeries.html]power series[/url] [img width=96,height=17]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline15.gif[/img] 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 [img width=62,height=17]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline16.gif[/img] does not tell anything about convergence of [img width=83,height=23]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline17.gif[/img]. Suppose that [img width=68,height=17]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline18.gif[/img] converges for some [img width=60,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline19.gif[/img], that each [img width=31,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline20.gif[/img] is differentiable on [img width=32,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline21.gif[/img], and that [img width=83,height=23]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline22.gif[/img] converges uniformly on [img width=32,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline23.gif[/img]. Then [img width=62,height=17]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline24.gif[/img] converges uniformly on [img width=32,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline25.gif[/img] to a function [img width=9,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline26.gif[/img], and for each [img width=54,height=15]https://mathworld.wolfram.com/images/equations/UniformConvergence/Inline27.gif[/img],[br][br][table][tr][td][img width=150,height=44]https://mathworld.wolfram.com/images/equations/UniformConvergence/NumberedEquation5.gif[/img][br][br][/td][/tr][/table]