(a) Let (an) be an increasing sequence of real numbers bounded from above and (bn) be a decreasing sequence. (i) Is (anbn) necessarily convergent? Justify your answer. (ii) Prove that if in addition bn ≥ 1 for all n EN then the sequence (anb¹) is convergent. (b) Let (an) be the sequence defined recursively by a1 = √3, an = 2an-1 + 3, n > 2. Prove that (an) is bounded. (ii) Prove that (an) is increasing. (iii) Making use of (i) and (ii) prove that the sequence (an) is convergent and compute its limit.
(a) Let (an) be an increasing sequence of real numbers bounded from above and (bn) be a decreasing sequence. (i) Is (anbn) necessarily convergent? Justify your answer. (ii) Prove that if in addition bn ≥ 1 for all n EN then the sequence (anb¹) is convergent. (b) Let (an) be the sequence defined recursively by a1 = √3, an = 2an-1 + 3, n > 2. Prove that (an) is bounded. (ii) Prove that (an) is increasing. (iii) Making use of (i) and (ii) prove that the sequence (an) is convergent and compute its limit.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(a) Let (an) be an increasing sequence of real numbers bounded from above and (bn)
be a decreasing sequence.
(i) Is (anbn) necessarily convergent? Justify your answer.
(ii) Prove that if in addition bn ≥ 1 for all n EN then the sequence (anb¹) is
convergent.
(b) Let (an) be the sequence defined recursively by
a1 = √√3,
an =
/2an-1 +3,
(i) Prove that (an) is bounded.
(ii) Prove that (an) is increasing.
n> 2.
(iii) Making use of (i) and (ii) prove that the sequence (an) is convergent and
compute its limit.
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[5]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1dd36e4-bdc9-4085-8f76-bccd4db94add%2F40543235-8f17-4463-bcf0-1ccfddd74c17%2Fsjfsc59_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Let (an) be an increasing sequence of real numbers bounded from above and (bn)
be a decreasing sequence.
(i) Is (anbn) necessarily convergent? Justify your answer.
(ii) Prove that if in addition bn ≥ 1 for all n EN then the sequence (anb¹) is
convergent.
(b) Let (an) be the sequence defined recursively by
a1 = √√3,
an =
/2an-1 +3,
(i) Prove that (an) is bounded.
(ii) Prove that (an) is increasing.
n> 2.
(iii) Making use of (i) and (ii) prove that the sequence (an) is convergent and
compute its limit.
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[5]
[5]
[5]
[5]
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