2. Let an be the sequence defined inductively by a₁ = 2 and an+1 (a) Prove by induction that an € [1, 2] for all n E N. (b) Prove that a2 ≥ 2 for all n € N. 2 = 1/2 (am + ²). an (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.
2. Let an be the sequence defined inductively by a₁ = 2 and an+1 (a) Prove by induction that an € [1, 2] for all n E N. (b) Prove that a2 ≥ 2 for all n € N. 2 = 1/2 (am + ²). an (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.
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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b,c and d please
![2. Let an be the sequence defined inductively by a₁ = 2 and an+1
(a) Prove by induction that an € [1, 2] for all n € N.
(b) Prove that a ≥ 2 for all n € N.
2
= 1/2 (a₁ + ²).
an
(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%2F7dd22a7c-6880-4a63-9133-40fb48f40bc4%2F9568eb60-70bc-4115-935d-bd2efa89609e%2Fv5uaat_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let an be the sequence defined inductively by a₁ = 2 and an+1
(a) Prove by induction that an € [1, 2] for all n € N.
(b) Prove that a ≥ 2 for all n € N.
2
= 1/2 (a₁ + ²).
an
(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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