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

appendix E Sigma Notation A35
2 Theorem If c is any constant (that is, it does not depend on i), then
(a) o n
ca i − c o n
a i
i−m
i−m
(c) o n
sa i 2 b i d − o n
a i 2 o n
b i
i−m
i−m i−m
(b) o n
sa i 1 b i d − o n
a i 1 o n
b i
i−m
i−m i−m
Proof To see why these rules are true, all we have to do is write both sides in
expanded form. Rule (a) is just the distributive property of real numbers:
ca m 1 ca m11 1 ∙ ∙ ∙ 1 ca n − csa m 1 a m11 1 ∙ ∙ ∙ 1 a n d
Rule (b) follows from the associative and commutative properties:
Rule (c) is proved similarly.
sa m 1 b m d 1 sa m11 1 b m11 d 1 ∙ ∙ ∙ 1 sa n 1 b n d
− sa m 1 a m11 1 ∙ ∙ ∙ 1 a n d 1 sb m 1 b m11 1 ∙ ∙ ∙ 1 b n d
■
EXAMPLE 3 Find o n
1.
SOLUTION
i−1
o n
1 − 1 1 1 1 ∙ ∙ ∙ 1 1 − n ■
i−1
n terms
EXAMPLE 4 Prove the formula for the sum of the first n positive integers:
o n
nsn 1 1d
i − 1 1 2 1 3 1 ∙ ∙ ∙ 1 n −
i−1
2
SOLUTIOn This formula can be proved by mathematical induction (see page 72) or
by the following method used by the German mathematician Karl Friedrich Gauss
(1777–1855) when he was ten years old.
Write the sum S twice, once in the usual order and once in reverse order:
S − 1 1 2 1 3 1 ∙ ∙ ∙ 1 sn 2 1d 1 n
S − n 1 sn 2 1d 1 sn 2 2d 1 ∙ ∙ ∙ 1 2 1 1
Adding all columns vertically, we get
2S − sn 1 1d 1 sn 1 1d 1 sn 1 1d 1 ∙ ∙ ∙ 1 sn 1 1d 1 sn 1 1d
On the right side there are n terms, each of which is n 1 1, so
2S − nsn 1 1d or S −
nsn 1 1d
2
■
EXAMPLE 5 Prove the formula for the sum of the squares of the first n positive
integers:
o n
nsn 1 1ds2n 1 1d
i 2 − 1 2 1 2 2 1 3 2 1 ∙ ∙ ∙ 1 n 2 −
i−1
6
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.