Consider the recursive sequence defined by 1 In+1 = In + 2 In Assume that 1 >0. Follow the steps below to prove that lim n = √2. 004-1 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. Solution: TYPE YOUR SOLUTION HERE! 2) If 0<<√√2, show that 22 > √√2. Using 1), that means the sequence is monotoni- cally decreasing and bounded below starting from n = 2. Solution: TYPE YOUR SOLUTION HERE! 3) Conclude that the sequence is convergent for all 1 >0. Show that for all ₁ > 0, the sequence converges to √2. Solution: TYPE YOUR SOLUTION HERE!

Consider the recursive sequence defined by 1 In+1 = In + 2 In Assume that 1 >0. Follow the steps below to prove that lim n = √2. 004-1 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. Solution: TYPE YOUR SOLUTION HERE! 2) If 0<<√√2, show that 22 > √√2. Using 1), that means the sequence is monotoni- cally decreasing and bounded below starting from n = 2. Solution: TYPE YOUR SOLUTION HERE! 3) Conclude that the sequence is convergent for all 1 >0. Show that for all ₁ > 0, the sequence converges to √2. Solution: TYPE YOUR SOLUTION HERE!

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.6: Inequalities

Problem 80E

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