1. Use the limit theorems to prove that the following sequences converge. sin(2n + 1). n+1 n+2 n an 15n²-6 5n21' bn 1 Cn=n- 2. Let an be the sequence defined inductively by a₁ = n; dn 2 and an+1 (a) Prove by induction that an € [1, 2] for all n € N. (b) Prove that a ≥2 for all n € N. 9n (n+8)!' an + an (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (by (a)) so it follows, by the mono- tone convergence theorem, that (an) converges to some limit L. Show that L > 0 and L² = 2.
1. Use the limit theorems to prove that the following sequences converge. sin(2n + 1). n+1 n+2 n an 15n²-6 5n21' bn 1 Cn=n- 2. Let an be the sequence defined inductively by a₁ = n; dn 2 and an+1 (a) Prove by induction that an € [1, 2] for all n € N. (b) Prove that a ≥2 for all n € N. 9n (n+8)!' an + an (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (by (a)) so it follows, by the mono- tone convergence theorem, that (an) converges to some limit L. Show that L > 0 and L² = 2.
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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![B Homework Problems
1. Use the limit theorems to prove that the following sequences converge.
an =
15n²-6
5n²1
bn
=
sin(2n +1)
n
Cn=n-
n+1
2+2
(a) Prove by induction that an € [1,2] for all n € N.
(b) Prove that a ≥2 for all n E N.
|n; dn =
2. Let an be the sequence defined inductively by a₁ = 2 and an+1 =
9n
(n+8)!'
-1/2 (a₁ + ²/2 ).
an
(c) Hence prove that the sequence is decreasing.
(d) We already know that (an) is bounded below (by (a)) so it follows, by the mono-
tone convergence theorem, that (an) converges to some limit L. Show that L > 0
and L² = 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed029406-a1c1-473f-a3a0-6fd0fbd8e89d%2F8e92bfa8-d919-42a5-be1f-ae1ca0d1fbcb%2F4iaayra_processed.jpeg&w=3840&q=75)
Transcribed Image Text:B Homework Problems
1. Use the limit theorems to prove that the following sequences converge.
an =
15n²-6
5n²1
bn
=
sin(2n +1)
n
Cn=n-
n+1
2+2
(a) Prove by induction that an € [1,2] for all n € N.
(b) Prove that a ≥2 for all n E N.
|n; dn =
2. Let an be the sequence defined inductively by a₁ = 2 and an+1 =
9n
(n+8)!'
-1/2 (a₁ + ²/2 ).
an
(c) Hence prove that the sequence is decreasing.
(d) We already know that (an) is bounded below (by (a)) so it follows, by the mono-
tone convergence theorem, that (an) converges to some limit L. Show that L > 0
and L² = 2.
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