and this becomes
| ƒ(x) = ƒ(a) + Σ∞r=1 | (x − a)r | ƒ(r)(a), |
| r! |
provided that (α) a positive number k can be found so that at all points in the interval between a and a + k (except these points) ƒ(x) has continuous differential coefficients of all finite orders, and at a has progressive differential coefficients of all finite orders; (β) Cauchy’s form of the remainder Rn, viz. [(x − a) / (n − 1)!] (1 − θ)n−1 ƒn {a + θ(x − a)}, has the limit zero when n increases indefinitely, for all values of θ between 0 and 1, and for all values of x in the interval between a and a + k, except possibly a + k. When these conditions are satisfied, the series (1) represents the function at all points of the interval between a and a + k, except possibly a + k, and the function is “analytic” (§ 13) in this domain. Obvious modifications admit of extension to an interval between a and a − k, or between a − k and a + k. When a series of the form (1) represents a function it is called “the Taylor’s series for the function.”
Taylor’s series is a power series, i.e. a series of the form
Σ∞n=0 an (x − a)n.
As regards power series we have the following theorems:
1. If the power series converges at any point except a there is a number k which has the property that the series converges absolutely in the interval between a − k and a + k, with the possible exception of one or both end-points.
2. The power series represents a continuous function in its domain of convergence (the end-points may have to be excluded).
3. This function is analytic in the domain, and the power series representing it is the Taylor’s series for the function.
The theory of power series has been developed chiefly from the point of view of the theory of functions of complex variables.