**Problem Statement:**1. Suppose \(\{x_n\}\) is a sequence of real numbers that is (i) increasing (that is, \(x_n \leq x_{n+1}\) for all \(n\)) and (ii) bounded above (that is, there exists some \(M \in \mathbb{R}\) so that \(x_n \leq M\) for all \(n\)). Prove that the sequence \(x_n\) converges.

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Answered: 1. Suppose {Tn} is a sequence of real…

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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Suppose that {an}_₁ is a sequence defined by an = n=1 the Squeezing Theorem. We wish to find sequences {bn}_₁ and {n}_₁ so that the following argument is correct: n=1 n=1 Which of the following definitions Select one alternative: n=1 "For each n ≥ 1, bn ≤ an ≤ Cn, and the sequences {bn}_₁ and {cn}_₁ both converge to the same limit. Therefore {an}_1 also converges to that limit." Sn=1 {bn}x=1 Obn Obn Obn - = = en-1 en+1 and Cn −1+ en en + 1 O bn = 1 and Cn = : 1. and cn = 1. −(−1)n + en en + 1 - and Cn (−1)n + en en + 1 1- en en and that we wish to show it converges using 1 {n}x=1 ake the argument correct
Let (an) and (bn) be sequences of real numbers such that (an) converges and (bn) diverges to +∞. Prove that (an + bn) diverges to +∞.
Suppose that the sequence x0, x₁, x2... is defined by xo = 3, x₁ = 0, x₂ = 5, and xk+3 = 13xk+1+12xê for k≥0. Find a general formula for x. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k. xk = 0
1
4. Consider a sequence aj = 3 and an+1 = v V12 + an for each n > 1. (a) by 4. Show by induction that {an} is monotone increasing and bounded above
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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.
4. 225
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The purpose of this exercise is to establish the convergence of an iterative process for approximating square roots. Define a sequence recursively as follows: X1 = 2 1 2 (xn-1 + 2 Xn Xn-1 A) Show that xn 2 v2 B) Show that the sequence is decreasing C) Show that the sequence converges to V2 D) Let ɛ X, - V2 and show that ɛn+1 which would mean letting o = 2v2 which 2xn makes ɛn+1 < o()2n (n = 1,2,3, ..) E) Generalize this process to find a sequence that converges to Va where x is any positive real number F) Generalize part (D) WITHOUT ANY PROOF and use it with x1 = 2 and a = 3 to determine E1 , ... , E6 Note: this will show how rapid the convergence of the recursion formula is in calculating square roots
Prove {(8n-5)/n} n=1 to infinity, is a Cauchy sequence in R with the usual metric dR(x,y) = abs(x-y)
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1. Suppose {Tn} is a sequence of real numbers that is (i) increasing (that is, In < In+1 for all n) and (ii) bounded above (that is, there exists some M ER so that rn < M for all n). Prove that the sequence rn Converges.