1.Σ1Consider the sequence ak =Vk +1- Vkand the infinite seriesa. Which of the followingVk+1+vkk=1is true?(A) lim akO andak converges.k=1(В)lim ar + 0 andak converges.k=1(C) lim akO andak diverges.k=1(D) lim ak 7 0 andak diverges.'k=1

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Answered: 1. Vk +1- Vk = 1 Consider the sequence…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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2 3 Let x be the sequence X 1, ,2, 3, 3' 4 = Select ALL of the correct statements (note that there may be more than one correct statement). O This sequence is convergent to None of these. This sequence is divergent, since it is unbounded. This seqeunce has a convergent subsequence. This sequence is divergent, since it has two subsequences with different limits.
Lemma 4. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {z„} e So for n > N. Then: (i) (1– p)zn N; (ii) {zn¢n} is increasing for all n > N. Proof. Assume that {yn} is a positive solution of equation (1.1) with the corres- ponding sequence {zn} E So. Then z, is positive, Zn 2 yn, and Yn = Zn – Pnyo(n) 2 (1– p)zn, n>N> no, so (i) is proved. It easy to see that z, E So implies lim b,(Az,)“ = 0; n00 otherwise we would eventually have Az, > 0 contradicting z, E So. Similarly, lim a„A(b,(Azn)“) = 0. A summation of equation (1.1) then yields 9szs+1 Zn+1 LIs. s=n s=n s=n Summing once more, we obtain as t=s s=n or Azn 2-Zn+1Qn. Hence, A(zn&n) = PnAzn + Zn+1Ao, > Zn+1(A0n – OnQn) =0 since {0,} is a solution of the difference equation (Ao, – Qnºn) = 0. Therefore, {z,9n} is increasing and this completes the proof of the lemma.
Mark each sequence {n} having the property that lim infan> inf{n: n € N}. n+∞0 U U 0 U Xn = = {cos(nπ/3)} (-1)"n n+1 Xn = In = 1 Xn= n 1 when n is odd, and n = 1+ - when n is even. n xn = (-1)" n n- (-1)"
peremine 2) an and a. for anmmeric sequence α₁=x, α₁₂=X13 WE **

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1. Vk +1- Vk = 1 Consider the sequence ak = and the infinite series ak. Which of the following Vk+1+vk k=1 is true? (A) lim ap O and ak converges. k=1 (В) lim ak + 0 and ak converges. k=1 (C) lim ak O and ak diverges. k=1 (D) lim a 7 0 and ak diverges. k=1