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

762 Chapter 11 Infinite Sequences and Series
In trying to show that lim n l ` R n sxd − 0 for a specific function f, we usually use the
following theorem.
9 Taylor’s Inequality If | f sn11d sxd | < M for | x 2 a |
R n sxd of the Taylor series satisfies the inequality
| R nsxd | < M
sn 1 1d! | x 2 a | n11
To see why this is true for n − 1, we assume that | f 0sxd |
f 0sxd < M, so for a < x < a 1 d we have
y x
a
f 0std dt < y x
M dt
a
< d, then the remainder
for | x 2 a | < d
< M. In particular, we have
Formulas for the
Taylor Remainder Term
As alternatives to Taylor’s Inequality,
we have the following formulas for the
remainder term. If f sn11d is continuous
on an interval I and x [ I, then
R nsxd − 1 n! yx sx 2 td n f sn11d std dt
a
This is called the integral form of the
remainder term. Another formula,
called Lagrange’s form of the remainder
term, states that there is a number z
between x and a such that
R nsxd − f sn11d szd
sx 2 adn11
sn 1 1d!
This version is an extension of the
Mean Value Theorem (which is the
case n − 0).
Proofs of these formulas, together
with discussions of how to use them to
solve the examples of Sections 11.10
and 11.11, are given on the website
www.stewartcalculus.com
Click on Additional Topics and then on
Formulas for the Remainder Term in
Taylor series.
An antiderivative of f 0 is f 9, so by Part 2 of the Fundamental Theorem of Calculus, we
have
Thus
f 9sxd 2 f 9sad < Msx 2 ad or f 9sxd < f 9sad 1 Msx 2 ad
y x
a
f 9std dt < y x
f f 9sad 1 Mst 2 adg dt
a
f sxd 2 f sad < f 9sadsx 2 ad 1 M
f sxd 2 f sad 2 f 9sadsx 2 ad < M 2
sx 2 ad2
But R 1 sxd − f sxd 2 T 1 sxd − f sxd 2 f sad 2 f 9sadsx 2 ad. So
R 1 sxd < M 2
sx 2 ad2
A similar argument, using f 0sxd > 2M, shows that
R 1 sxd > 2 M 2
sx 2 ad2
So | R 1sxd | < M 2 |x 2 a | 2
sx 2 ad2
2
Although we have assumed that x . a, similar calculations show that this inequality is
also true for x , a.
This proves Taylor’s Inequality for the case where n − 1. The result for any n is
proved in a similar way by integrating n 1 1 times. (See Exercise 83 for the case n − 2.)
Note In Section 11.11 we will explore the use of Taylor’s Inequality in approximating
functions. Our immediate use of it is in conjunction with Theorem 8.
In applying Theorems 8 and 9 it is often helpful to make use of the following fact.
10
x n
lim − 0 for every real number x
nl` n!
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