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

A38
appendix E Sigma Notation
Exercises
1–10 Write the sum in expanded form.
1. o 5
i−1
3. o 6
i−4
5. o 4
k−0
7. o n
i 10
i−1
si 2. o 6
3 i 4. o 6
2k 2 1
2k 1 1
n21
9. o s21d j
j−0
i−1
6. o 8
i 3
i−4
x k
k−5
8. o n13
10. o n
11–20 Write the sum in sigma notation.
11. 1 1 2 1 3 1 4 1 ∙ ∙ ∙ 1 10
12. s3 1 s4 1 s5 1 s6 1 s7
13.
14.
1
2 1 2 3 1 3 4 1 4 5 1 ∙ ∙ ∙ 1 19
20
3
7 1 4 8 1 5 9 1 6 10 1 ∙ ∙ ∙ 1 23
27
15. 2 1 4 1 6 1 8 1 ∙ ∙ ∙ 1 2n
16. 1 1 3 1 5 1 7 1 ∙ ∙ ∙ 1 s2n 2 1d
17. 1 1 2 1 4 1 8 1 16 1 32
18.
1
1 1 1 4 1 1 9 1 1 16 1 1 25 1 1 36
19. x 1 x 2 1 x 3 1 ∙ ∙ ∙ 1 x n
20. 1 2 x 1 x 2 2 x 3 1 ∙ ∙ ∙ 1 s21d n x n
21–35 Find the value of the sum.
21. o 8
i−4
23. o 6
j−1
25. o 20
n−1
27. o 4
i−0
29. o n
i−1
31. o n
i−1
33. o n
i−1
s3i 2 2d 22. o 6
3 j11 24. o 8
j 2
j−n
i−1
i−3
k−0
s21d n 26. o 100
4
i−1
s2 i 1 i 2 d 28. o 4
2i
si 2 1 3i 1 4d
si 1 1dsi 1 2d
30. o n
1
i 1 1
f sx id Dx i
isi 1 2d
cos k
2 32i
i−22
i−1
32. o n
i−1
34. o n
i−1
s2 2 5id
s3 1 2id 2
isi 1 1dsi 1 2d
35. o n
i−1
si 3 2 i 2 2d
36. Find the number n such that o n
i−1
37. Prove formula (b) of Theorem 3.
i − 78.
38. Prove formula (e) of Theorem 3 using mathematical
induction.
39. Prove formula (e) of Theorem 3 using a method similar to
that of Example 5, Solution 1 [start with s1 1 id 4 2 i 4 g.
40. Prove formula (e) of Theorem 3 using the following method
published by Abu Bekr Mohammed ibn Alhusain Alkarchi
in about ad 1010. The figure shows a square ABCD in
which sides AB and AD have been divided into segments of
lengths 1, 2, 3, . . . , n. Thus the side of the square has length
nsn 1 1dy2 so the area is fnsn 1 1dy2g 2 . But the area is also
the sum of the areas of the n “gnomons” G 1, G 2, . . . , G n
shown in the figure. Show that the area of G i is i 3 and conclude
that formula (e) is true.
D
n
.
5
G∞
.
.
.
G n
4 G¢
3 G£
2
A 1 G
12 3 4 5 . . . n B
41. Evaluate each telescoping sum.
(a) o n
fi 4 2 si 2 1d 4 g (b) o 100
s5 i 2 5 i21 d
i−1
i−1
(c)
i−3S o 99 1 2 1
(d)
i i 1 1D o n
sa i 2 a i21d
i−1
42. Prove the generalized triangle inequality:
< o n
43–46 Find the limit.
43. lim
n l ` o n 1
i−1
S i 2
n nD
45. lim
nl ` o n 2
FS 2i
i−1 n
3
nD
Zo n
a i Z
i−1
1 5S nDG
2i
i−1 | ai |
C
44. lim
n l ` o n 1
FS i 3
1
i−1 n nD 1G
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