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

Section 11.2 Series 711
(b) What is the concentration after the nth dose?
(c) What is the limiting concentration?
Solution
(a) Just before the daily dose of medication is administered, the concentration is
reduced to 30% of the preceding day’s concentration, that is, 0.3C n . With the new dose,
the concentration is increased by 0.2 mgymL and so
C n11 − 0.2 1 0.3C n
Starting with C 0 − 0 and putting n − 0, 1, 2 into this equation, we get
C 1 − 0.2 1 0.3C 0 − 0.2
C 2 − 0.2 1 0.3C 1 − 0.2 1 0.2s0.3d − 0.26
C 3 − 0.2 1 0.3C 2 − 0.2 1 0.2s0.3d 1 0.2s0.3d 2 − 0.278
The concentration after three days is 0.278 mgymL.
(b) After the nth dose the concentration is
C n − 0.2 1 0.2s0.3d 1 0.2s0.3d 2 1 . . . 1 0.2s0.3d n21
This is a finite geometric series with a − 0.2 and r − 0.3, so by Formula 3 we have
C n − 0.2 f1 2 s0.3dn g
1 2 0.3
− 2 7 f1 2 s0.3dn g mgymL
(c) Because 0.3 , 1, we know that lim nl` s0.3d n − 0. So the limiting concentration is
2
lim
nl` Cn − lim
nl` 7 f1 2 s0.3dn g − 2 7 s1 2 0d − 2 7 mgymL n
Example 6 Write the number 2.317 − 2.3171717. . . as a ratio of integers.
SOLUTION
2.3171717. . . − 2.3 1 17
10 3 1 17
10 5 1 17
10 7 1 ∙ ∙ ∙
After the first term we have a geometric series with a − 17y10 3 and r − 1y10 2 .
Therefore
2.317 − 2.3 1
17
10 3
1 2 1
10 2 − 2.3 1
− 23
10 1 17
990 − 1147
495
17
1000
99
100
n
Example 7 Find the sum of the series ò x n , where | x | , 1.
n−0
Solution Notice that this series starts with n − 0 and so the first term is x 0 − 1.
(With series, we adopt the convention that x 0 − 1 even when x − 0.)
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