Thomas Calculus 13th [Solutions]

Section 5.2 Sigma Notation and Limits of Finite Sums 351
29. (a)
k
7
3 3(7) 21
1
(c) Let j k 2 k j 2; if k 3
500
(b) 7 7(500) 3500
k 1
264 262
j 1 and if k 264 j 262 10 10 10(262) 2620
k 3 j 1
30. (a) Let j k 8 k j 8; if k 9 j 1 and if k 36 j 28
36 28 28 28
k ( j 8) j 8
k 9 j 1 j 1 j 1
28(28 1)
8(28) 630
2
17 15
2 2
(b) Let j k 2 k j 2; if k 3 j 1 and if k 17 j 15 k ( j 2)
k 3 j 1
15 15 15 15
2 2
( j 4 j 4) j 4 j 15(15 1)(2(15) 1) 15(15 1)
6 2
j 1 j 1 j 1 j 1
4(15) 1240 480 60 1780
71
(c) Let j k 17 k j 17; if k 18 j 1 and if k 71 j 54 k( k 1)
k 3
54 54
54 54 54
2 2
( j 17)(( j 17) 1) ( j 33 j 272) j 33 j 272
j 1 j 1
j 1 j 1 j 1
54(54 1)(2(54) 1) 54(54 1)
33
6 2
272(54) 53955 49005 14688 117648
31. (a)
(c)
n
4 4n
k 1
n n n
n( n 1)
2
( k 1) k 1 n n n
2 2
k 1 k 1 k 1
(b)
n
c
k 1
cn
n
2
32. (a) 1 2n 1 2n n 1 2n (b)
n n
k 1
n
(c) k 1 n( n 1) n 1
2 2 2 2n
k 1 n n
n c c
n n
k 1
n
c
33. (a) (b) (c)
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