1 = 2 and an+1 2 2. Let an be the sequence defined inductively by a1 An + An (a) Prove by induction that an E [1, 2] for all n E N. (b) Prove that a, > 2 for all n e N. (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (by (a)) so it follows, by the mono- tone convergence theorem, that (an) converges to some limit L. Show that L> 0 and L? = 2.
1 = 2 and an+1 2 2. Let an be the sequence defined inductively by a1 An + An (a) Prove by induction that an E [1, 2] for all n E N. (b) Prove that a, > 2 for all n e N. (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (by (a)) so it follows, by the mono- tone convergence theorem, that (an) converges to some limit L. Show that L> 0 and L? = 2.
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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Question
![1
2 and an+1
2
An +
An
2. Let an be the sequence defined inductively by a1 =
(a) Prove by induction that an E [1, 2] for all n E N.
(b) Prove that a, > 2 for all n e N.
(c) Hence prove that the sequence is decreasing.
(d) We already know that (an) is bounded below (by (a)) so it follows, by the mono-
tone convergence theorem, that (an) converges to some limit L. Show that L> 0
and L2 = 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F986f6c25-9d5d-4e3f-93bc-35b073c4428b%2F6d36724c-9fa2-445b-95e5-8d23a1c453e8%2Fae557x_processed.png&w=3840&q=75)
Transcribed Image Text:1
2 and an+1
2
An +
An
2. Let an be the sequence defined inductively by a1 =
(a) Prove by induction that an E [1, 2] for all n E N.
(b) Prove that a, > 2 for all n e N.
(c) Hence prove that the sequence is decreasing.
(d) We already know that (an) is bounded below (by (a)) so it follows, by the mono-
tone convergence theorem, that (an) converges to some limit L. Show that L> 0
and L2 = 2.
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