11.8 Power Series
Many familiar (and unfamiliar) functions can be written in the form
cc
as an infinite sum of the product of certain numbers
c
and powers of the variable
.
23
01
2
3
xc
x
++ + +
L
x
n
Such expressions are called
power series with center 0;
the numbers
c
are called its
coefficients,
.
n
0
n
n
n
cx
∞
=
∑
Slightly more general, an expression of the form
is called a
power series in (xa)
or a
power series centered at a,
.
2
2
()()
xac
xa
+−
+−+
L
0
()
n
n
n
cxa
∞
=
−
∑
So, the question becomes, "when does the power series converge?" Any of the series tests are
available for use, but most often the Ratio Test is used. In general this will boil down to
1
lim
n
n
n
c
c
+
→∞
−
.
When this limit is between –1 and 1, the series converges.
There are only 3 possibilities for how this series can converge.
Theorem (page 742)
(i)
The series converges only when x=a.
(ii)
The series converges for all x.
(iii)
There is a positive number
R
such that the series converges if xa<R and diverges
if xa>R.
R is called the radius of convergence.
Note that the special cases of xa=R need to be checked separately. If the series only
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 Fall '08
 Estrada
 Power Series, Mathematical Series, lim, Radius of convergence

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