22. Uniform Convergence.—We shall suppose that the domain of convergence of an infinite series of functions is an interval with the possible exception of isolated points. Let ƒ(x) be the sum of the series at any point x of the domain, and ƒn(x) the sum of the first n + 1 terms. The condition of convergence at a point a is that, after any positive number ε, however small, has been specified, it must be possible to find a number n so that |ƒm(a) − ƒp(a)| < ε for all values of m and p which exceed n. The sum, ƒ(a), is the limit of the sequence of numbers ƒn(a) at n = ∞. The convergence is said to be “uniform” in an interval if, after specification of ε, the same number n suffices at all points of the interval to make |ƒ(x) − ƒm(x)| < ε for all values of m which exceed n. The numbers n corresponding to any ε, however small, are all finite, but, when ε is less than some fixed finite number, they may have an infinite superior limit (§ 7); when this is the case there must be at least one point, a, of the interval which has the property that, whatever number N we take, ε can be taken so small that, at some point in the neighbourhood of a, n must be taken > N to make |ƒ(x) − fm(x)| < ε when m > n; then the series does not converge uniformly in the neighbourhood of a. The distinction may be otherwise expressed thus: Choose a first and ε afterwards, then the number n is finite; choose ε first and allow a to vary, then the number n becomes a function of a, which may tend to become infinite, or may remain below a fixed number; if such a fixed number exists, however small ε may be, the convergence is uniform.
For example, the series sin x − ½ sin 2x + 1⁄3 sin 3x − ... is convergent for all real values of x, and, when π > x > −π its sum is ½x; but, when x is but a little less than π, the number of terms which must be taken in order to bring the sum at all near to the value of ½x is very large, and this number tends to increase indefinitely as x approaches π. This series does not converge uniformly in the neighbourhood of x = π. Another example is afforded by the series
| Σ∞n=0 | nx | − | (n + 1)x | , |
| n²x² + 1 | (n + 1)²x² + 1 |
of which the remainder after n terms is nx/(n²x² + 1). If we put x = 1/n, for any value of n, however great, the remainder is ½; and the number of terms required to be taken to make the remainder tend to zero depends upon the value of x when x is near to zero—it must, in fact, be large compared with 1/x. The series does not converge uniformly in the neighbourhood of x = 0.
As regards series whose terms represent continuous functions we have the following theorems:
(1) If the series converges uniformly in an interval it represents a function which is continuous throughout the interval.
(2) If the series represents a function which is discontinuous in an interval it cannot converge uniformly in the interval.
(3) A series which does not converge uniformly in an interval may nevertheless represent a function which is continuous throughout the interval.
(4) A power series converges uniformly in any interval contained within its domain of convergence, the end-points being excluded.
(5) If Σ∞r=0 ƒr(x) = ƒ(x) converges uniformly in the interval between a and b