Thomas Calculus 13th [Solutions]

Section 10.2 Infinite Series 719
87. Since the sum of a finite number of terms is finite, adding or subtracting a finite number of terms from a series
that diverges does not change the divergence of the series.
88. Let An
a1 a2
a n and lim An
A . Assume an b n converges to S. Let
n
Sn a1 b1 a2 b2 an bn Sn a1 a2 an b1 b2
bn
b1 b2 bn Sn An lim b1 b2
bn S A b n converges. This contradicts the
n
assumption that b n diverges; therefore, an b n diverges.
89. (a)
(b)
2 5 2 1 r r 3 ; 2 2 3 2 3
1 r 5 5 5 5
13
2
13 3 13 13 3 13 3
2
13 3
3
5 1 r r ;
1 r 10 10 2 2 10 2 10 2 10
2
90.
91.
b 2b 1 1 b b
1 e e 9 1 e e 8 b ln 8
b
1 e 9 9 9
2 3 4 5 2n
2n
1
sn
1 2r r 2r r 2r r 2 r , n 0, 1,
2 4 2n 3 5 2n 1
s 1 2 2 2 2 lim 1 2r 1 2r
n n r r r r r r r s
,
2 2 2
n 1 r 1 r 1 r
2
if r 1 or r 1
92. area 2 2
2
2 2 (1) 2 1 4 2 1 1 4 2
2 1
8 m 1
2
( 0.12)(24)
93. (a) After 24 hours, before the second pill: 300e
16.840 mg; after 48 hours, the amount present
after 24 hours continues to decay and the dose taken at 24 hours has 24 hours to decay, so the amount
( 0.12)(48) ( 0.12)(24)
present is 300e
300e
0.945 16.840 17.785 mg.
2
(b) The long-run quantity of the drug is
( 0.12)(24)
300 ( 0.12)(24)
n
300 e
e
( 0.12)(24)
17.84 mg.
1
1 e
94.
a 1 r n
L s a
ar
n 1 r 1 r 1 r
n
95. (a) The endpoint of any closed interval remaining at any stage of the construction will remain in the Cantor
set, so some points in the set include 0,
1
,
2
,
1
,
2
,
7
,
8
,
1
,
2
,
7
,
8
,1.
(b) The lengths of the intervals removed are:
1
Stage 1:
3
Stage 2: 1 1
1 2
3 3 9
Stage 3: 1 1 2 4
1
3 3 9 27
and so on.
27 27 9 9 27 27 3 3 9 9
Thus the sum of the lengths of the intervals removed is
n 1
1 2 1 1
3 3 3 1 (2 / 3)
n 1
1.
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