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Let f be differentiable on R with a = sup{|f'(x)| : x ≤ R} < 1. = f(so), $2 = f($1), etc. Prove (Sn) is a = (a) Select So E R and define Sn = f(Sn-1) for n ≥ 1. Thus 8₁ convergent sequence. Hint: To show that (sn) is Cauchy, first show |Sn+1 - Sn| ≤ a|sn - Sn-1| for n ≥ 1. (b) Show f has a fixed point, i.e., f(s) = s for some s in R.

Let f be differentiable on R with a = sup{|f'(x)| : x ≤ R} < 1. = f(so), $2 = f($1), etc. Prove (Sn) is a = (a) Select So E R and define Sn = f(Sn-1) for n ≥ 1. Thus 8₁ convergent sequence. Hint: To show that (sn) is Cauchy, first show |Sn+1 - Sn| ≤ a|sn - Sn-1| for n ≥ 1. (b) Show f has a fixed point, i.e., f(s) = s for some s in R.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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Q7 Let f be differentiable on R with a = sup{|f'(x)| : x = R} <1. (a) Select so E R and define Sn = f($n-1) for n ≥ 1. Thus 81 = f($0), $2 = f($₁), etc. Prove (Sn) is a = convergent sequence. Hint: To show that (sn) is Cauchy, first show |Sn+1 - Sn| ≤ a sn - Sn-1| for n ≥ 1. (b) Show f has a fixed point, i.e., f(s) = s for some s in R.
Transcribed Image Text:Q7 Let f be differentiable on R with a = sup{|f'(x)| : x = R} <1. (a) Select so E R and define Sn = f($n-1) for n ≥ 1. Thus 81 = f($0), $2 = f($₁), etc. Prove (Sn) is a = convergent sequence. Hint: To show that (sn) is Cauchy, first show |Sn+1 - Sn| ≤ a sn - Sn-1| for n ≥ 1. (b) Show f has a fixed point, i.e., f(s) = s for some s in R.
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