James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 11.10 Taylor and Maclaurin Series 761
The conclusion we can draw from Theorem 5 and Example 1 is that if e x has a power
series expansion at 0, then
x
e x − ò
n
n−0 n!
So how can we determine whether e x does have a power series representation?
Let’s investigate the more general question: under what circumstances is a function
equal to the sum of its Taylor series? In other words, if f has derivatives of all orders,
when is it true that
f sxd − ò
n−0
f snd sad
n!
sx 2 ad n
As with any convergent series, this means that f sxd is the limit of the sequence of partial
sums. In the case of the Taylor series, the partial sums are
T n sxd − o n
i−0
f sid sad
i!
sx 2 ad i
− f sad 1 f 9sad
1!
sx 2 ad 1 f 0sad
2!
sx 2 ad 2 1 ∙ ∙ ∙ 1 f snd sad
n!
sx 2 ad n
y=T(x)
y=T£(x)
FIGURE 1
y
(0, 1)
y=´
y=T£(x)
y=T(x)
y=T¡(x)
0 x
As n increases, T nsxd appears to
approach e x in Figure 1. This suggests
that e x is equal to the sum of its Taylor
series.
Notice that T n is a polynomial of degree n called the nth-degree Taylor polynomial of
f at a. For instance, for the exponential function f sxd − e x , the result of Example 1 shows
that the Taylor polynomials at 0 (or Maclaurin polynomials) with n − 1, 2, and 3 are
T 1 sxd − 1 1 x T 2 sxd − 1 1 x 1 x 2
2!
T 3 sxd − 1 1 x 1 x 2
2! 1 x 3
3!
The graphs of the exponential function and these three Taylor polynomials are drawn in
Figure 1.
In general, f sxd is the sum of its Taylor series if
If we let
f sxd − lim
nl`
T n sxd
R n sxd − f sxd 2 T n sxd so that f sxd − T n sxd 1 R n sxd
then R n sxd is called the remainder of the Taylor series. If we can somehow show that
lim n l ` R n sxd − 0, then it follows that
lim Tnsxd − lim f f sxd 2 R n sxdg − f sxd 2 lim R n sxd − f sxd
nl` nl` nl`
We have therefore proved the following theorem.
8 Theorem If f sxd − T n sxd 1 R n sxd, where T n is the nth-degree Taylor polynomial
of f at a and
lim Rnsxd − 0
nl`
for | x 2 a |
| x 2 a | , R.
, R, then f is equal to the sum of its Taylor series on the interval
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