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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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part c and part d (already asked for other parts)
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04625ac1-43ff-4999-b93a-55388fc0c5e2%2F0c28b104-2489-47cf-a43d-187aa03304c1%2Fo3eyhfj_processed.png&w=3840&q=75)
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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