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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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84. If \( \Sigma a_n \) is divergent and \( c \neq 0 \), show that \( \Sigma c a_n \) is divergent.
85. If \( \Sigma a_n \) is convergent and \( \Sigma b_n \) is divergent, show that the series \( \Sigma (a_n + b_n) \) is divergent.
*[Hint: Argue by contradiction.]*
86. If \( \Sigma a_n \) and \( \Sigma b_n \) are both divergent, is \( \Sigma (a_n + b_n) \) necessarily divergent?
87. Suppose that a series \( \Sigma a_n \) has positive terms and its partial sums \( s_n \) satisfy the inequality \( s_n \leq 1000 \) for all \( n \). Explain why
\( \Sigma a_n \) must be convergent.
88. The Fibonacci sequence was defined in Section 11.1 by the equations
\( f_1 = 1, \quad f_2 = 1, \quad f_n = f_{n-1} + f_{n-2} \quad \text{for} \quad n \geq 3 \)
Show that each of the following statements is true:
(a) \( \frac{1}{f_{n-1}f_n} = \frac{1}{f_{n-1}f_n} - \frac{1}{f_nf_{n+1}} \)
(b) \( \sum_{n=2}^{\infty} \frac{1}{f_{n-1}f_{n+1}} = 1 \)
(c) \( \sum_{n=2}^{\infty} \frac{f_n}{f_{n-1}f_{n+1}} = 2 \)
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 \(\left(\frac{1}{3}, \frac{2}{3}\right)\). That leaves the two intervals \(\left[0, \frac{1}{3}\right]\](/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:Here is a transcription and explanation for the educational content:
**Text Transcription:**
84. If \( \Sigma a_n \) is divergent and \( c \neq 0 \), show that \( \Sigma c a_n \) is divergent.
85. If \( \Sigma a_n \) is convergent and \( \Sigma b_n \) is divergent, show that the series \( \Sigma (a_n + b_n) \) is divergent.
*[Hint: Argue by contradiction.]*
86. If \( \Sigma a_n \) and \( \Sigma b_n \) are both divergent, is \( \Sigma (a_n + b_n) \) necessarily divergent?
87. Suppose that a series \( \Sigma a_n \) has positive terms and its partial sums \( s_n \) satisfy the inequality \( s_n \leq 1000 \) for all \( n \). Explain why
\( \Sigma a_n \) must be convergent.
88. The Fibonacci sequence was defined in Section 11.1 by the equations
\( f_1 = 1, \quad f_2 = 1, \quad f_n = f_{n-1} + f_{n-2} \quad \text{for} \quad n \geq 3 \)
Show that each of the following statements is true:
(a) \( \frac{1}{f_{n-1}f_n} = \frac{1}{f_{n-1}f_n} - \frac{1}{f_nf_{n+1}} \)
(b) \( \sum_{n=2}^{\infty} \frac{1}{f_{n-1}f_{n+1}} = 1 \)
(c) \( \sum_{n=2}^{\infty} \frac{f_n}{f_{n-1}f_{n+1}} = 2 \)
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 \(\left(\frac{1}{3}, \frac{2}{3}\right)\). That leaves the two intervals \(\left[0, \frac{1}{3}\right]\
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