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!
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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Transcribed Image Text: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!
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