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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 73E
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