2. Let an be the sequence defined inductively by a₁ = 2 and an+1 == 1/2 (a₁ + ²2² ). an (a) Prove by induction that an € [1, 2] for all n € N. (b) Prove that az > 2 for all n N (hint: write a < 2 and find a contradiction). (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (from part (a)) so it follows by the monotone convergence theorem (or at least, it will follow from your answer to question 3 below you are allowed to assume it here), that (an) converges to some limit L. Show that L is the positive real number such that L² = 2 (hint: 0 = L for some L. How can we use this we are saying we know we can say lim an on two sides of an equation). n→∞
2. Let an be the sequence defined inductively by a₁ = 2 and an+1 == 1/2 (a₁ + ²2² ). an (a) Prove by induction that an € [1, 2] for all n € N. (b) Prove that az > 2 for all n N (hint: write a < 2 and find a contradiction). (c) Hence prove that the sequence is decreasing. (d) We already know that (an) is bounded below (from part (a)) so it follows by the monotone convergence theorem (or at least, it will follow from your answer to question 3 below you are allowed to assume it here), that (an) converges to some limit L. Show that L is the positive real number such that L² = 2 (hint: 0 = L for some L. How can we use this we are saying we know we can say lim an on two sides of an equation). n→∞
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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Question
![2. Let an be the sequence defined inductively by a₁ = 2 and an+1 ==
1/2 (a₁ + ²2² ).
an
(a) Prove by induction that an € [1, 2] for all n € N.
(b) Prove that az > 2 for all n N (hint: write a < 2 and find a contradiction).
(c) Hence prove that the sequence is decreasing.
(d) We already know that (an) is bounded below (from part (a)) so it follows by the
monotone convergence theorem (or at least, it will follow from your answer to
question 3 below you are allowed to assume it here), that (an) converges to
some limit L. Show that L is the positive real number such that L² = 2 (hint:
0
=
L for some L. How can we use this
we are saying we know we can say lim an
on two sides of an equation).
n→∞](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7f27c73-756c-4a47-8ea7-79b047cfb8fe%2F5adb10fb-1e08-426d-88e2-3d61f50dcb67%2Fk5z7ooe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let an be the sequence defined inductively by a₁ = 2 and an+1 ==
1/2 (a₁ + ²2² ).
an
(a) Prove by induction that an € [1, 2] for all n € N.
(b) Prove that az > 2 for all n N (hint: write a < 2 and find a contradiction).
(c) Hence prove that the sequence is decreasing.
(d) We already know that (an) is bounded below (from part (a)) so it follows by the
monotone convergence theorem (or at least, it will follow from your answer to
question 3 below you are allowed to assume it here), that (an) converges to
some limit L. Show that L is the positive real number such that L² = 2 (hint:
0
=
L for some L. How can we use this
we are saying we know we can say lim an
on two sides of an equation).
n→∞
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