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The class of functions that we know are continuous includes, among others, the polynomials, the rational functions, and the th root functions. We can combine these functions in various ways, e.g., sums, products, quotients, and so on.We also can combine continuous functions using composition, so that we know that th roots of rational functions are also continuous. The set of all functions obtained in this manner is calledthe class of “algebraic functions.” Now that we also have developed a notion of limit,or infinite sum, we can construct other continuous functions.
We introduce next a new kind of function. It is a natural generalization of a polynomial function.Among these will be the exponential function and the trigonometric functions.We begin by discussing functions of a complex varible, although totally analogous definitions and theorems hold for functions of a real variable.
Let be a sequence of real or complex numbers. By the power series function we mean the function where the domain is the set of all for which the infinite series converges, and where is the rule that assigns to such a the sum of the series.
The numbers defining a power series function are called the coefficients of the function.
We associate to a power series function its sequence of partial sums. We write
Notice that polynomial functions are very special cases of power series functions. They are the power series functions for which the coefficients are all 0 beyond some point. Note also that each partial sum for any power series function is itself a polynomial function of degree less than or equal to Moreover, if is a power series function, then for each in its domain we have Evidently, every power series function is a “limit” of a sequence of polynomials.
Obviously, the domain of a power series function depends on the coefficients determining the function. Our first goal is to describe this domain.
Let be a power series function: with domain Then:
Part (1) is clear.
To see part 2, assume that belongs to and that We wish to show that the infinites series converges. In fact, we will show that is convergent, i.e., that is absolutely convergent. We are given that the infinite series converges, which implies that the terms tend to 0. Hence, let be a number such that for all and set Then and therefore the infinite series is convergent. Finally, which, by the Comparison Test, implies that is convergent, as desired.
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