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Uniform Convergence

A Series $\sum_{n=1}^\infty u_n(x)$ is uniformly convergent to $S(x)$ for a set $E$ of values of $x$ if, for each $\epsilon>0$, an Integer $N$ can be found such that

\vert S_n(x)-S(x)\vert < \epsilon
\end{displaymath} (1)

for $n\geq N$ and all $x\in E$. 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
1. The series sum
f(x)=\sum_{n=1}^\infty u_n(x)
\end{displaymath} (2)

is continuous,

2. The series may be integrated term by term
\int_a^b f(x)\,dx = \sum_{n=1}^\infty \int_a^b u_n(x)\,dx,
\end{displaymath} (3)


3. The series may be differentiated term by term
{d\over dx} f(x) = \sum_{n=1}^\infty {d\over dx} u_n(x).
\end{displaymath} (4)

See also Abel's Theorem, Abel's Uniform Convergence Test, Weierstraß M-Test


Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 299-301, 1985.

© 1996-9 Eric W. Weisstein