88. The Fibonacci sequence was defined in Section 11.1 by the equations fi = 1, f2= 1, fn= fn=1 + fn-2 n> 3 %3D %3D Show that each of the following statements is true. 1 1 (a) fn-1 fnt1 1 fn-1 fn fn fa+1 00 1 00 (b) E 3D1 n=2 fn-1 fn+1 fa (c) %3| n=2 fn-1 fn+1
88. The Fibonacci sequence was defined in Section 11.1 by the equations fi = 1, f2= 1, fn= fn=1 + fn-2 n> 3 %3D %3D Show that each of the following statements is true. 1 1 (a) fn-1 fnt1 1 fn-1 fn fn fa+1 00 1 00 (b) E 3D1 n=2 fn-1 fn+1 fa (c) %3| n=2 fn-1 fn+1
Chapter9: Sequences, Probability And Counting Theory
Section9.4: Series And Their Notations
Problem 10TI: Determine whether the sum of the infinite series is defined. 24+(12)+6+(3)+
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#88
![84. If E a, is divergent and c 0, show that E ca, is divergent.
85. If E a, is convergent and E b, is divergent, show that the series
(an + bn) is divergent. [Hint: Argue by contradiction.]
nat
Ci;
n
86. If E a, and E b, are both divergent, is E (a, + b„) necessarily
divergent?
of
hus
87. Suppose that a series E a, has positive terms and its partial
sums Sn satisfy the inequality Sn < 1000 for all n. Explain why
Ea, must be convergent.
88. The Fibonacci sequence was defined in Section 11.1 by the
equations
fi = 1, f2= 1, fn= fn=1 + fn-2
n> 3
Show that each of the following statements is true.
1
1
(a)
fn-1 fa+1
fn-1 fn fn fn+1
1
fn
(c) Σ
n=2 fn-1 fn+1
T
Σ
= 1
n=2 fn-1 fn+1
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
interval
(G,). That leaves the two intervals 0, and , 1 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
open](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcba78722-b839-46bb-b46d-8bfd06aff38d%2Fc9e94a19-ad47-47ee-afb7-9e823b8e8f4b%2F3two7qwz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:84. If E a, is divergent and c 0, show that E ca, is divergent.
85. If E a, is convergent and E b, is divergent, show that the series
(an + bn) is divergent. [Hint: Argue by contradiction.]
nat
Ci;
n
86. If E a, and E b, are both divergent, is E (a, + b„) necessarily
divergent?
of
hus
87. Suppose that a series E a, has positive terms and its partial
sums Sn satisfy the inequality Sn < 1000 for all n. Explain why
Ea, must be convergent.
88. The Fibonacci sequence was defined in Section 11.1 by the
equations
fi = 1, f2= 1, fn= fn=1 + fn-2
n> 3
Show that each of the following statements is true.
1
1
(a)
fn-1 fa+1
fn-1 fn fn fn+1
1
fn
(c) Σ
n=2 fn-1 fn+1
T
Σ
= 1
n=2 fn-1 fn+1
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
interval
(G,). That leaves the two intervals 0, and , 1 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
open
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