Advanced High-School Mathematics

SECTION 5.3 Concept of Power Series 283

a+ar+ar^2 +··· =

a

1 −r

.

Let’s make a minor cosmetic change: rather than writingr in the
above sum, we shall writex:

a+ax+ax^2 +··· =

a

1 −x

, |x|< 1.

In other words, if we set

f(x) = a+ax+ax^2 +··· =

∑∞

n=0

axn and set g(x) =

a

1 −x

,

then the following facts emerge:

(a) The domain off is− 1 < x <1, and the domain ofgisx 6 = 1.

(b)f(x) =g(x) for allxin the interval− 1 < x <1.

We say, therefore, that

∑∞

n=0

axn is thepower series representation

ofg(x),valid on the interval− 1 < x <1.

So what is a power series anyway? Well, it’s just an expression of
the form

∑∞

n=0

anxn,

wherea 0 , a 1 , a 2 , ...are just real constants. For any particular value of
xthis infinite summay or may not converge; we’ll have plenty to
say about issues of convergence.