3.6 Power series functions

The class of functions that we know are continuous includes, among others, the polynomials, the rational functions, and the $n$ 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 $n$ 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 ${\left\{a_n\right\}}_0^{\infty }$ be a sequence of real or complex numbers. By the power series function $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ we mean the function $f:S\to C$ where the domain $S$ is the set of all $z\in C$ for which the infinite series $\sum a_n{z}^n$ converges, and where $f$ is the rule that assigns to such a $z\in S$ the sum of the series.

The numbers $\left\{a_n\right\}$ defining a power series function are called the coefficients of the function.

We associate to a power series function $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ its sequence $\left\{S_N\right\}$ of partial sums. We write

$$S_N\left(z\right)=\sum _{n=0}^N{a}_n{z}^n.$$

Notice that polynomial functions are very special cases of power series functions. They are the power series functions for which the coefficients $\left\{a_n\right\}$ are all 0 beyond some point. Note also that each partial sum $S_N$ for any power series function is itself a polynomial function of degree less than or equal to $N.$ Moreover, if $f$ is a power series function, then for each $z$ in its domain we have $f\left(z\right)={lim}_N{S}_N\left(z\right).$ Evidently, every power series function is a “limit” of a sequence of polynomials.

Obviously, the domain $S\equiv S_f$ of a power series function $f$ depends on the coefficients $\left\{a_n\right\}$ determining the function. Our first goal is to describe this domain.

Let $f$ be a power series function: $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ with domain $S.$ Then:

1.  0 belongs to $S.$
2. If a number $t$ belongs to $S,$ then every number $u,$ for which $\left|u\right|<\left|t\right|,$ also belongs to $S.$
3.   $S$ is a disk of radius $r$ around 0 in $C$ (possibly open, possibly closed, possibly neither, possibly infinite). That is, $S$ consists of the disk $B_r\left(0\right)=\{z:|z|<r\}$ possibly together with some of the points $z$ for which $\left|z\right|=r.$
4. The radius $r$ of the disk in part (3) is given by the Cauchy-Hadamard formula:
   $$r=\frac{1}{lim\; sup|a_n{|}^{1/n}},$$
   which we interpret to imply that $r=0$ if and only if the limsup on the right is infinite, and $r=\infty$ if and only if that limsup is $0.$

Part (1) is clear.

To see part 2, assume that $t$ belongs to $S$ and that $\left|u\right|<\left|t\right|.$ We wish to show that the infinites series $\sum a_n{u}^n$ converges. In fact, we will show that $\sum |a_n{u}^n|$ is convergent, i.e., that $\sum a_n{u}^n$ is absolutely convergent. We are given that the infinite series $\sum a_n{t}^n$ converges, which implies that the terms $a_n{t}^n$ tend to 0. Hence, let $B$ be a number such that $|a_n{z}^n|\le B$ for all $n,$ and set $\alpha =\left|u\right|/\left|t\right|.$ Then $\alpha <1,$ and therefore the infinite series $\sum B{\alpha }^n$ is convergent. Finally, $|a_n{u}^n|=|a_n{t}^n|{\alpha }^n\le B{\alpha }^n,$ which, by the Comparison Test, implies that $\sum |a_n{u}^n|$ is convergent, as desired.