Thus a power series in x/x0 will define a function of x for all x in the interval in which the series converges. In terms of the summation notation,

we can write the power series as

X n0

an(xx0) n ,

and if the function to which this infinite series converges is denoted by f(x),

often called its sum function, we may write

f (x) X n0

an(xx0) n:

The interval in which this power series will converge will depend on the

coefficients an , the point (number) x0 about which the series is expanded and x itself. To determine the convergence properties, let us apply the ratio test to

the power series, in which the nth term is an (x/x0) n and the (n/1)th term is

an1(xx0) n1:

Then we know from the ratio test that the power series will converge if

lim n0 j an1(x x0)n1an(x x0)n jB1,

which is equivalent to

lim n0

j an1an j jxx0 j B1

lim n0

j n1an j


Assuming that jan1/an j is always defined, that is an"/0, we find after division that

jxx0jB lim n0 j anan1 j:

If we now define the number r to be

r lim n0 j anan1 j,

we see that the power series converges absolutely (the ratio test is a test for

absolute convergence) if

jxx0 jBr: This inequality defines the interval


in which the power series is absolutely convergent. The number r is called the

radius of convergence of the power series, and the interval x0/r B/x B/x0/r is called the interval of convergence. As the ratio test fails when L/1, it is not possible to say on the basis of the above argument whether the series con-

verges at the end points of the interval of convergence.