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

718 Chapter 11 Infinite Sequences and Series
79. The figure shows two circles C and D of radius 1 that touch
at P. The line T is a common tangent line; C 1 is the circle that
touches C, D, and T; C 2 is the circle that touches C, D, and C 1;
C 3 is the circle that touches C, D, and C 2. This procedure can
be continued indefinitely and produces an infinite sequence of
circles hC nj. Find an expression for the diameter of C n and thus
provide another geometric demonstration of Example 8.
C
C£
1
C
1
80. A right triangle ABC is given with /A − and | AC | − b.
CD is drawn perpendicular to AB, DE is drawn perpendicular
to BC, EF AB, and this process is continued indefi nitely,
as shown in the figure. Find the total length of all the
perpendiculars
P
C¡
| CD | 1 | DE | 1 | EF | 1 | FG | 1 ∙ ∙ ∙
in terms of b and .
B
81. What is wrong with the following calculation?
H
0 − 0 1 0 1 0 1 ∙ ∙ ∙
− s1 2 1d 1 s1 2 1d 1 s1 2 1d 1 ∙ ∙ ∙
F
G
D
E
¨
C
A
b
D
T
84. If o a n is divergent and c ± 0, show that o ca n is divergent.
85. If o a n is convergent and o b n is divergent, show that the series
o sa n 1 b nd is divergent. [Hint: Argue by contradiction.]
86. If o a n and o b n are both divergent, is o sa n 1 b nd necessarily
divergent?
87. Suppose that a series o a n has positive terms and its partial
sums s n satisfy the inequality s n < 1000 for all n. Explain why
o a n must be convergent.
88. The Fibonacci sequence was defined in Section 11.1 by the
equations
f 1 − 1, f 2 − 1, f n − f n21 1 f n22 n > 3
Show that each of the following statements is true.
1
(a) − 1 2 1
f n21 f n11 f n21 f n f n f n11
(b) ò
n−2
1
f n21 f n11
− 1
(c) ò
n−2
f n
f n21 f n11
− 2
89. The Cantor set, named after the German mathematician
Georg Cantor (1845–1918), is constructed as follows. We start
with the closed interval [0, 1] and remove the open interval
( 1 3 , 2 3 ). That leaves the two intervals f0, 1 3 g and f 2 3 , 1g and we
remove the open middle third of each. Four intervals remain
and again we remove the open middle third of each of them.
We continue this procedure indefinitely, at each step removing
the open middle third of every interval that remains from the
preceding step. The Cantor set consists of the numbers that
remain in [0, 1] after all those intervals have been removed.
(a) Show that the total length of all the intervals that are
removed is 1. Despite that, the Cantor set contains infinitely
many numbers. Give examples of some numbers in
the Cantor set.
(b) The Sierpinski carpet is a two-dimensional counterpart
of the Cantor set. It is constructed by removing the center
one-ninth of a square of side 1, then removing the centers
of the eight smaller remaining squares, and so on. (The
figure shows the first three steps of the construction.)
Show that the sum of the areas of the removed squares
is 1. This implies that the Sierpinski carpet has area 0.
− 1 2 1 1 1 2 1 1 1 2 1 1 ∙ ∙ ∙
− 1 1 s21 1 1d 1 s21 1 1d 1 s21 1 1d 1 ∙ ∙ ∙
− 1 1 0 1 0 1 0 1 ∙ ∙ ∙ − 1
(Guido Ubaldus thought that this proved the existence of God
because “something has been created out of nothing.”)
82. Suppose that o ǹ−1 an san ± 0d is known to be a convergent
series. Prove that oǹ−1 1yan is a divergent series.
83. Prove part (i) of Theorem 8.
90. (a) A sequence ha nj is defined recursively by the equation
a n − 1 2 san21 1 an22d for n > 3, where a1 and a2 can be
any real numbers. Experiment with various values of a 1
and a 2 and use your calculator to guess the limit of the
sequence.
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