ABSTRACT
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
jxx0jB1:
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
x0rBxBx0r
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.
