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Consider the recursive sequence defined by In+1 - ½ (+212) Assume that ₁ >0. Follow the steps below to prove that lim n = √2. 11-00 1) If ₁ =√√2, show that the sequence is constant. If ₁>√√2, show that the sequence is monotonically decreasing and bounded below by √2. 2) If 0<<√√2, show that 2 > √√2. Using 1), that means the sequence is monotoni- cally decreasing and bounded below starting from n = 2. 3) Conclude that the sequence is convergent for all ₁ >0. Show that for all ₁ > 0, the sequence converges to √2.

Consider the recursive sequence defined by In+1 - ½ (+212) Assume that ₁ >0. Follow the steps below to prove that lim n = √2. 11-00 1) If ₁ =√√2, show that the sequence is constant. If ₁>√√2, show that the sequence is monotonically decreasing and bounded below by √2. 2) If 0<<√√2, show that 2 > √√2. Using 1), that means the sequence is monotoni- cally decreasing and bounded below starting from n = 2. 3) Conclude that the sequence is convergent for all ₁ >0. Show that for all ₁ > 0, the sequence converges to √2.

Advanced Engineering Mathematics
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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Consider the recursive sequence defined by In+1 - ½ (+212) Assume that ₁ >0. Follow the steps below to prove that lim n = √2. 11-00 1) If ₁ =√√2, show that the sequence is constant. If ₁>√√2, show that the sequence is monotonically decreasing and bounded below by √2. 2) If 0<<√√2, show that 2 > √√2. Using 1), that means the sequence is monotoni- cally decreasing and bounded below starting from n = 2. 3) Conclude that the sequence is convergent for all ₁ >0. Show that for all ₁ > 0, the sequence converges to √2.
Transcribed Image Text:Consider the recursive sequence defined by In+1 - ½ (+212) Assume that ₁ >0. Follow the steps below to prove that lim n = √2. 11-00 1) If ₁ =√√2, show that the sequence is constant. If ₁>√√2, show that the sequence is monotonically decreasing and bounded below by √2. 2) If 0<<√√2, show that 2 > √√2. Using 1), that means the sequence is monotoni- cally decreasing and bounded below starting from n = 2. 3) Conclude that the sequence is convergent for all ₁ >0. Show that for all ₁ > 0, the sequence converges to √2.
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