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Fix a constant ≤ K≤. Let (n)-1 be a sequence such that r₁ = K and for all n 9 Fn+1 = n(1 − Fn). (a) Prove that for all n we have ≤ n ≤ 10 (b) Prove that for all n we have [1-In-In+1 <. (c) Verify the identity 9 [²ªn+1(1 − In+1)] – [²In(1 − In)] = ² (Fn+1 – Fn)(1 – In – In+1). - - - (d) Prove that for all n 9 |£n+2-£n+1|≤|£n+1 − En|- 10

Fix a constant ≤ K≤. Let (n)-1 be a sequence such that r₁ = K and for all n 9 Fn+1 = n(1 − Fn). (a) Prove that for all n we have ≤ n ≤ 10 (b) Prove that for all n we have [1-In-In+1 <. (c) Verify the identity 9 [²ªn+1(1 − In+1)] – [²In(1 − In)] = ² (Fn+1 – Fn)(1 – In – In+1). - - - (d) Prove that for all n 9 |£n+2-£n+1|≤|£n+1 − En|- 10

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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2. CAUCHY SEQUENCES Fix a constant ≤K≤7. Let (n)-1 be a sequence such that x₁ = K and for all n 9 In+1 = (a) Prove that for all n we have ≤ n ≤ 10 (b) Prove that for all n we have [1-In-In+1 ≤ ². (c) Verify the identity 9 [²7Tn+1(1 − En+1)] − [In(1 - In)] = ² (En+1 − En) (1 - En — Fn+1). - (d) Prove that for all n -(1-₂). q®(1−2n). (e) Prove that for all n we have | #n+2=In+1| ≤ 10|²n+1 - In|- and therefore that |In+2 - Xn+1| ≤ ( )" |T2 - 11]. 10 (f) Prove that for all 1 <m <n we have m-1 m |£n − xm| ≤ [( ² )™¯¹ + ( ² )™. +... + 10. n-3 + n (²) ²²2₂-1₁ m 1 Inm≤ 10( 10 (™¹|2 - ₁1. (g) Prove that (n)-1 is Cauchy and find what it converges to.
Transcribed Image Text:2. CAUCHY SEQUENCES Fix a constant ≤K≤7. Let (n)-1 be a sequence such that x₁ = K and for all n 9 In+1 = (a) Prove that for all n we have ≤ n ≤ 10 (b) Prove that for all n we have [1-In-In+1 ≤ ². (c) Verify the identity 9 [²7Tn+1(1 − En+1)] − [In(1 - In)] = ² (En+1 − En) (1 - En — Fn+1). - (d) Prove that for all n -(1-₂). q®(1−2n). (e) Prove that for all n we have | #n+2=In+1| ≤ 10|²n+1 - In|- and therefore that |In+2 - Xn+1| ≤ ( )" |T2 - 11]. 10 (f) Prove that for all 1 <m <n we have m-1 m |£n − xm| ≤ [( ² )™¯¹ + ( ² )™. +... + 10. n-3 + n (²) ²²2₂-1₁ m 1 Inm≤ 10( 10 (™¹|2 - ₁1. (g) Prove that (n)-1 is Cauchy and find what it converges to.
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