Uniform convergence

The ideas of previous article can be extended to the case where the u_n are functions of x denoted by u_n(x). In such case the sequences or series will converge of diverge according to the particular value of x. The set of values of x for which a sequence or series converges is called the region of convergence, denoted .

The series u₁(x) + u₂(x) + … converges to the sum S(x) in a region if given ε > 0 there exists a number N, which in general depends on both ε and x, such that |S(x) – S_n(x)| N where S_n(x) = u₁(x) + … + u_n(x). If you can find N depending only on ε and not on x, the series converges uniformly to S(x) in . Uniformly convergent series have many important advantages as indicated in the following theorems.

1. If u_n(x), n = 1, 2, 3, … are continuous in a ≤ x ≤ b and ∑ u_n(x) is uniformly convergent to S(x) in a ≤ x ≤ b, then S(x) is continuous in a ≤ x ≤ b.
2. If ∑u(x) converges uniformly to S(x) in a ≤ x ≤ b and u_n(x), n = 1, 2, 3, … are integrable in a ≤ x ≤ b, then
   
3. If u_n(x), n = 1, 2, 3, … are continuous and have continuous derivatives in a ≤ x ≤ b and if ∑u_n(x) converges to S(x) while ∑u’_n(x) is uniformly convergent in a ≤ x ≤ b, then
   
4. If there is aset of positive constants M_n, n = 1, 2, 3, … such that |u_n| ≤ M_n in and ∑M_n converges, then ∑u_n(x) is uniformly convergent [and also absolutely convergent] in .

An important test for uniform convergence, often called the Weierstrass M test, is given by the above.