2. Let an be the sequence defined inductively by ai = 2 and an+1 = 1 an + 2 an (a) Prove by induction that an € [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.
2. Let an be the sequence defined inductively by ai = 2 and an+1 = 1 an + 2 an (a) Prove by induction that an € [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.
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
![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 a² ≥ 2 for all n € N.
= 1/2 (a₁ +²2²)
an
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%2F69cce4ac-4bf6-4e6b-8636-bf160e045b58%2F82383d0d-7ef2-4f7b-acc0-de27aed7af65%2Fey8x7f_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 a² ≥ 2 for all n € N.
= 1/2 (a₁ +²2²)
an
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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