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

appendix E Sigma Notation A37
We list the results of Examples 3, 4, and 5 together with a similar result for cubes (see
Exercises 37–40) as Theorem 3. These formulas are needed for finding areas and evaluating
integrals in Chapter 5.
3 Theorem Let c be a constant and n a positive integer. Then
(a) o n
1 − n (b) o n
c − nc
i−1
i−1
(c) o n
i−1
(e) o n
i−1
nsn 1 1d
i −
2
nsn 1 1d
i −F G
3
2
2
(d) o n
i−1
i 2 −
nsn 1 1ds2n 1 1d
6
The type of calculation in Example 7
arises in Chapter 5 when we compute
areas.
EXAMPLE 6 Evaluate o n
i−1
is4i 2 2 3d.
SOLUTIOn Using Theorems 2 and 3, we have
EXAMPLE 7 Find lim
n l ` o n
i−1
SOLUTION
lim
n l `
o n
is4i 2 2 3d − o n
s4i 3 2 3id − 4 o n
i 3 2 3 o n
i
i−1
i−1
i−1
i−1
2
nsn 1 1d nsn 1 1d
− 4F G 2 3
2 2
−
nsn 1 1df2nsn 1 1d 2 3g
2
− nsn 1 1ds2n 2 1 2n 2 3d
2
3
FS i 2
1 1G.
n nD
3
FS i 2
1 on
i−1 n nD 1G − lim
n l ` o n
− lim
n l `F
− lim
n l `F
− lim
n l `F
− lim
n l `F
i−1F 3 n 3 i 2 1 3 nG
1G
3 o n
i 2 1 3 n 3 i−1 n on
i−1
3 nsn 1 1ds2n 1 1d
3
1 G
3 n 6
n ? n
S DS 1 2 ? n n ? n 1 1 2n 1 1
n n
D 1 3G
1
2 ? 1 S1 1 1 1
nDS2 nD 1 1 3G
− 1 2 ? 1 ? 1 ? 2 1 3 − 4 ■
■
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