Let (an)n∈ℕ be a row in ℝ such that a0=3 and an+1= (an)/2 + 2/(an) for n≥0 (a) Proof that an≥2 for all n∈ℕ First show that (x/2)+(2/x)≥2 for all x≥2. Then apply the induction on n. (b) Proof that an+1≤an for all n∈ℕ. (c) Prove that (an) is convergent and calculate a= limn→∞an

Let (an)n∈ℕ be a row in ℝ such that a0=3 and an+1= (an)/2 + 2/(an) for n≥0 (a) Proof that an≥2 for all n∈ℕ First show that (x/2)+(2/x)≥2 for all x≥2. Then apply the induction on n. (b) Proof that an+1≤an for all n∈ℕ. (c) Prove that (an) is convergent and calculate a= limn→∞an

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

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Let (a_n)_n∈ℕ be a row in ℝ such that a₀=3 and

a_n+1= (a_n)/2 + 2/(a_n) for n≥0

(a) Proof that a_n≥2 for all n∈ℕ

First show that (x/2)+(2/x)≥2 for all x≥2. Then apply the induction on n.

(b) Proof that a_n+1≤a_nfor all n∈ℕ.

_{Please if able provide the steps with some explanation, thank you in advance}

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