6.1 Power series and functions  (Page 25/8)

A power series is a type of series with terms involving a variable. More specifically, if the variable is x , then all the terms of the series involve powers of x . As a result, a power series can be thought of as an infinite polynomial. Power series are used to represent common functions and also to define new functions. In this section we define power series and show how to determine when a power series converges and when it diverges. We also show how to represent certain functions using power series.

Form of a power series

A series of the form

where x is a variable and the coefficients c _n are constants, is known as a power series    . The series

is an example of a power series. Since this series is a geometric series with ratio $r=\left|x\right|,$ we know that it converges if $\left|x\right|<1$ and diverges if $\left|x\right|\ge 1.$

To make this definition precise, we stipulate that $x^0=1$ and ${\left(x-a\right)}^0=1$ even when $x=0$ and $x=a,$ respectively.

The series

and

are both power series centered at $x=0.$ The series

is a power series centered at $x=2.$

Convergence of a power series

Since the terms in a power series involve a variable x , the series may converge for certain values of x and diverge for other values of x . For a power series centered at $x=a,$ the value of the series at $x=a$ is given by $c_0.$ Therefore, a power series always converges at its center. Some power series converge only at that value of x . Most power series, however, converge for more than one value of x . In that case, the power series either converges for all real numbers x or converges for all x in a finite interval. For example, the geometric series ${\displaystyle \sum _{n=0}^{\infty }{x}^n}$ converges for all x in the interval $\left(\mathrm{-1},1\right),$ but diverges for all x outside that interval. We now summarize these three possibilities for a general power series.

Proof

Suppose that the power series is centered at $a=0.$ (For a series centered at a value of a other than zero, the result follows by letting $y=x-a$ and considering the series ${\displaystyle \sum _{n=1}^{\infty }{c}_n{y}^n}.)$ We must first prove the following fact:

If there exists a real number $d\ne 0$ such that ${\displaystyle \sum _{n=0}^{\infty }{c}_n{d}^n}$ converges, then the series ${\displaystyle \sum _{n=0}^{\infty }{c}_n{x}^n}$ converges absolutely for all x such that $\left|x\right|<\left|d\right|.$

Since ${\displaystyle \sum _{n=0}^{\infty }{c}_n{d}^n}$ converges, the n th term $c_n{d}^n\to 0$ as $n\to \infty .$ Therefore, there exists an integer N such that $\left|c_n{d}^n\right|\le 1$ for all $n\ge N.$ Writing