+1(3) Define a sequence by $₁ = = 0, and Sn+1 = 8+¹ for all n € N.3(a) Prove by induction that 0 ≤ Sn ≤ 1 and s2 + 1 ≥ 3sn for all n.(b) Show that (sn) converges.(c) What is lim sn? Prove your answer using limit theorems.

Answered: Image /qna-images/answer/7624d215-940b-43c6-bb5e-b31ce11cecb1.jpg

Answered: (3) Define a sequence by $1 = 0, and…

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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Q#4 Consider the arithmetic sequence S, where the i-th term of S is defined as follows for positive integer į: S() = 2 + 6*(i-1) So S(1) = 2, S(2) = 8, S(3) = 14, and so on. We will use induction to prove the closed form of the sum of the first k terms of S is equal to 3k^2 - k, for every positive integer k. 1. Show your base case(s). 2. Show the proof of your inductive step. Be explicit about how you are using your inductive hypothesis and the claim you want to prove for k.
14
Consider the recursively defined sequence (Hn)n>1 where 1. H₁ = 1 2. Hn+1 = Hn + n²1 for n ≥ 1 Compare the quantities H₂n+1 and H₂n + 1/2 for n ≥ 0 by creating a table with small values of n. Prove your relation by induction.
1 2. Let an be the sequence defined inductively by a₁ = 2 and an+1 = (a) Prove by induction that an € [1, 2] for all n € N. (b) Prove that a ≥2 for all ne N. 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.
7. 1)Detine the sequence: an = n²-2n+1 (n=1,2,3, ...) recursively. 2)Define the sequence: an = (n+1)! (n= 1,2,3, ..) recursively. 3) Find the non-recursive formula for f (n): f (0) = 6, f (n) = f (n – 1) + 15 for n> 1 4) Find the non-recursive formula for f (n) :f (0) = 3, f (n) = -2f (n – 1)/7 for n>1
(1) A sequence an is given by a₁ = √2, an+1 = √2+ an. (a) By induction or otherwise, show that an is increasing and bounded above by 3. Apply the Monotonic Sequence Theorem to show that exists. (b) Find limn-+0 an.
2. Consider the sequence an given by 2n +1 n³ (a) Calculate the first four terms of the sequence (as integers or fractions). an = for n ≥ 1. (b) Is this sequence bounded above or below, or not bounded at all? Explain your answer in detail. Provide bounds if possible. (c) Does this sequence converge or diverge? If it converges, find the limit.

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(3) Define a sequence by $1 = 0, and Sn+1 = +¹ for all n € N. 3 (a) Prove by induction that 0 ≤ Sn ≤ 1 and s2 + 1 ≥ 3sn for all n. (b) Show that (sn) converges. (c) What is lim sn? Prove your answer using limit theorems.