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 £© Section
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 £© Power Series
We are now in a position to pick up the story we left off in Section : the extension of
Taylor polynomials to Taylor series. We shall see that a Taylor series is a type of infinite
series whose nth partial sum is a Taylor polynomial. Such series are examples of power
series, objects that we will study in this section before considering Taylor series in Section
.
Definition An infinite series of the form
∞
a (x − c)n = a + a (x − c) + a (x − c)2 + · · · ()
X n 0 1 2
n=0
is called a power series in x about c.
Example The infinite series
∞
xn x2 x3
= 1 + x + + + · · ·
X n! 2! 3!
n=0
is a power series in x about 0. Note that if we let
xn |x|n
bn = =
n! n!
for n = 0, 1, 2, . . ., then
|x|n+1
b (n + 1)! |x|
lim n+1 = lim = lim = 0
→∞→∞ n →∞
n bn n |x| n n + 1
n!
for any value of x. That is, by the ratio test, the series is absolutely convergent, and
hence convergent, for any value of x. Thus if we define a function, called the exponential
function, by
∞
xn
exp(x) = ()
X n!
n=0
then this function is defined for all values of x. We shall have much more to say about this
function, which may be thought of as the simplest “infinite” polynomial which is defined
for all real numbers, in Chapter 6.
1
2 Power Series Section
Notice that the convergence of () for all x implies, by the nth term test for diver-
gence, that
xn
lim = 0 ()
n→∞ n!
for any value of x. We have seen particular cases of this limit in the past, but this is the
first time we have had a simple proof that it is always 0.
Example Recall that the Taylor polynomial of order 2n + 1 for sin(x) at 0 is
n (−1)kx2k+1
P (x) = .
2n+1 X (2k + 1)!
k=0
Hence P2n+1(x) is a partial sum of the power series
∞
(−1)kx2k+1
. ()
X (2k + 1)!
k=0
If, for k = 0, 1, 2, . . ., we le