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

46.
lim
nl ` o n 3
FS1 1 3i
3
2 2S1 1
i−1 n nD
nDG
3i
47. Prove the formula for the sum of a finite geometric series with
first term a and common ratio r ± 1:
o n
ar i21 − a 1 ar 1 ar 2 1 ∙ ∙ ∙ 1 ar n21 − asr n 2 1d
i−1
r 2 1
48. Evaluate o n
i−1
49. Evaluate o n
i−1
3
2 i21 .
50. Evaluate
i−1F o m
o n
appendix F Proofs of Theorems A39
s2i 1 2 i d.
j−1
si 1 jdG.
In this appendix we present proofs of several theorems that are stated in the main body
of the text. The sections in which they occur are indicated in the margin.
Section 2.3
Limit Laws Suppose that c is a constant and the limits
lim f sxd − L and lim tsxd − M
x l a x l a
exist. Then
1. lim f f sxd 1 tsxdg − L 1 M
x l a
3. lim
x l a
fcf sxdg − cL
5. lim
x l a
f sxd
tsxd − L M if M ± 0
2. lim
x l a
f f sxd 2 tsxdg − L 2 M
4. lim
x l a
f f sxdtsxdg − LM
Proof of Law 4 Let « . 0 be given. We want to find . 0 such that
if 0 , | x 2 a | , then | f sxdtsxd 2 LM | , «
In order to get terms that contain | f sxd 2 L | and | tsxd 2 M | , we add and subtract
Ltsxd as follows:
| f sxdtsxd 2 LM | − | f sxdtsxd 2 Ltsxd 1 Ltsxd 2 LM |
− | f f sxd 2 Lg tsxd 1 Lftsxd 2 Mg |
< | f f sxd 2 Lg tsxd | 1 | Lftsxd 2 Mg |
− | f sxd 2 L | | tsxd | 1 | L | | tsxd 2 M |
(Triangle Inequality)
We want to make each of these terms less than «y2.
Since lim x l a tsxd − M, there is a number 1 . 0 such that
if 0 , | x 2 a | , 1 then | tsxd 2 M | , «
2(1 1 | L |)
Also, there is a number 2 . 0 such that if 0 , | x 2 a | , 2, then
| tsxd 2 M | , 1
and therefore
| tsxd | − | tsxd 2 M 1 M | < | tsxd 2 M | 1 | M | , 1 1 | M |
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.