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

$$\vert S_n(x)-S(x)\vert < \epsilon$$ (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

$$f(x)=\sum_{n=1}^\infty u_n(x)$$ (2)

is continuous,

2. The series may be integrated term by term

$$\int_a^b f(x)\,dx = \sum_{n=1}^\infty \int_a^b u_n(x)\,dx,$$ (3)

and

3. The series may be differentiated term by term

$${d\over dx} f(x) = \sum_{n=1}^\infty {d\over dx} u_n(x).$$ (4)

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

References

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