Identify the key steps needed to prove that a sequence , CR converges to ER in the limit n → ∞.Proof:A Let € < 0[Select]B Let € < 0[Select]C Let € > 0D Let € 20A Choose n E N so that< €B Choose NEN so that Vn > N:- x < €C Choose N € N so that Vn > N:N-Xn\ < €D Choose N E N so that VnN:n = x < €E Choose N € N so that Vn N!n = x < €F Choose NEN so that Vn > N:xnx > €E Let € = 0

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Answered: Identify the key steps needed to prove…

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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a) Suppose (an) is Cauchy and that for every k∈N, the interval (−1/k,1/k) contains atleast one term of (an). Can we say that (an) converges to 0? Either show that it does or give a counter-example. b) Suppose (xn) is Cauchy and (yn) is bounded and monotone. Can we say anything aboutwhether the sequence (xn yn) converges or not? Justify your answer!
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.
Write out the first five terms of the sequence with, In(n) determine whether the sequence n+1 n=1 converges, and if so find its limit. In(n) Enter the following information for an = n+1 · аз a4 a5 In(n) lim n→00 n +1 |||| ||
Exercise 9. Let {a,} be a bounded sequence and {bn}_1 be a sequence converges to 0. Prove that the sequence {an · bn}n=1 converges to 0.
Prove that a sequence {an} converges to 0 if and only if the sequence of absolute values 5 | an | 6 converges to 0.
Give an example of i) a Cauchy sequence which is not monotonic; ii) a monotonic sequence which is not Cauchy; iii) a bounded sequence which is not Cauchy; iv) Can you find an example of a Cauchy sequence which is unbounded? If not, why not? Explain your answers. For (i) take = xn= Show that {xn} is a Cauchy but not monotonic. For(ii), take= xn=n Show that{xn} is monotonic. Take &=;n E N; and n > N and m=n+1 Explain why (-1)" n |xn−Xn+1| > E Conclude that {xn) is monotonic but it is not Cauchy. For (iii), take xn = 1 +(-1)^. Show that {xn} is bounded, i.e. show that |xn| <2; for all n E N. Compute |xn-Xn+1| Explain why {xn} cannot be a Cauchy sequence. For (iv) Recall that every Cauchy sequence is bounded. Conclude that no such an example of a sequence {x} can be constructed.
Let a1 = 1 and an+1 = [1-1/(n+1)^2]an for all n ≥ 1. 1) Show that the sequence {an}n≥1 converges. 2) Find its limit
v 8 9. 10 Compute the sums below. (Assume that the terms in the first sum are consecutive terms of an arithmetic sequence.) 5+2 + (-1) + ... + (-295) = [] 123 E (-s1 + 12) = 0 į = 1 Save For Later Submi Check © 2021 McGraw Hill LLC. AlIl Rights Reserved. Terms of Use | Privacy Centem Results Per Page 12 v
Determine if the sequence is monotonic and if it is bounded. 2"5" an = n! nal Select the correct answer below and, if necessary, fill in the answer box(es) to complete your choice. OA. (a) is monotonic because the sequence is nonincreasing. The sequence has a least upper bound when n = but is unbounded because it has no lower bound. (Simplify your answer.) OB. (a) is not monotonic. The sequence is unbounded with no upper or lower bound. OC. (a) is not monotonic. The sequence is bounded by a greatest lower bound of when n. (Simplify your answers.) and a least upper bound OD. (a) is monotonic because the sequence is nondecreasing. The sequence has a greatest lower bound of but is unbounded because it has no upper bound. (Simplify your answer.)

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