Again it can be proved that, if ƒ(x) is continuous throughout a given interval, polynomials in x of finite degrees can be found, so as to form an infinite series of polynomials whose sum is equal to ƒ(x) at all points of the interval. Methods of representation of continuous functions by infinite series of rational fractional functions have also been devised.
Particular interest attaches to continuous functions which are not differentiable. Weierstrass gave as an example the function represented by the series Σ ∞0 an cos (bn xπ), where a is positive and less than unity, and b is an odd integer exceeding (1 + 3⁄2π)/a. It can be shown that this series is uniformly convergent in every interval, and that the continuous function ƒ(x) represented by it has the property that there is, in the neighbourhood of any point x0, an infinite aggregate of points x′, having x0 as a limiting point, for which {ƒ(x′) − ƒ(x0)} / (x′ − x0) tends to become infinite with one sign when x′ − x0 approaches zero through positive values, and infinite with the opposite sign when x′ − x0 approaches zero through negative values. Accordingly the function is not differentiable at any point. The definite integral of such a function ƒ(x) through the interval between a fixed point and a variable point x, is a continuous differentiable function F(x), for which F′(x) = ƒ(x); and, if ƒ(x) is one-signed throughout any interval F(x) is monotonous throughout that interval, but yet F(x) cannot be represented by a curve. In any interval, however small, the tangent would have to take the same direction for infinitely many points, and yet there is no interval in which the tangent has everywhere the same direction. Further, it can be shown that all functions which are everywhere continuous and nowhere differentiable are capable of representation by series of the form Σanφn(x), where Σan is an absolutely convergent series of numbers, and φn(x) is an analytic function whose absolute value never exceeds unity.
25. Calculations with Divergent Series.—When the series described in (1) and (2) of § 24 diverge, they may, nevertheless, be used for the approximate numerical calculation of the values of the function, provided the calculation is not carried beyond a certain number of terms. Expansions in series which have the property of representing a function approximately when the expansion is not carried too far are called “asymptotic expansions.” Sometimes they are called “semi-convergent series”; but this term is avoided in the best modern usage, because it is often used to describe series whose convergence depends upon the order of the terms, such as the series 1 − ½ + 1⁄3 − ...
In general, let ƒ0(x) + ƒ1(x) + ... be a series of functions which does not converge in a certain domain. It may happen that, if any number ε, however small, is first specified, a number n can afterwards be found so that, at a point a of the domain, the value ƒ(a) of a certain function ƒ(x) is connected with the sum of the first n + 1 terms of the series by the relation |ƒ(a) − Σ nr = 0 ƒr(a)| < ε. It must also happen that, if any number N, however great, is specified, a number n′(>n) can be found so that, for all values of m which exceed n′, |Σ mr = 0 ƒr(a)| > N. The divergent series ƒ0(x) + ƒ1(x) + ... is then an asymptotic expansion for the function f(x) in the domain.
The best known example of an asymptotic expansion is Stirling’s formula for n! when n is large, viz.
n! = √(2π) ½nn + ½ e−n + θ/12n,
where θ is some number lying between 0 and 1. This formula is included in the asymptotic expansion for the Gamma function. We have in fact
log {Γ(x)} = (x − ½) log x − x + ½ log 2π + ω(x),
where ω(x) is the function defined by the definite integral
ω(x) = ∫ ∞0 {(1 − e−t)−1 − t−1 − ½} t−1 e−tx dt.