Let an be a sequence of real numbers such that an+1 2 an, Vn 21 and an + a asn + 00. Use this preamble to answer questions 11 and 1211. Which of these statements best describe(s) a, as a Cauchy sequence?I. for any given e > 0, 3 n(e) E R Ə | an – a |< e Vn 2 n(e).II. for any given e > 0,| an – am < e V m, n 2 N, for some NE RIII. | an - am → 0 as m, n + 0A. I and II onlyB. I and III onlyС. П onlyD. III onlyE. None of the above choices A, B,C or D12. Which of the choices below give(s) the best definition of convergence of a,?I. for any given e > 0,3 n(e) E R Ə | an – a |< e V n 2 n(e).II. for any given e > 0, 3 n(e) E R Ə | a, – am |< e Vn z n(e).III. for any give e > 0, 3 n(e) e IR Ə a- e< an < a+ e Vn 2 n(e).A. I onlyB. I and III onlyС. П only-D. I, II and IIIE. None of the above choices A, B,C or D1 and rn+1Van + 2, V n 2 1. Let P, : n+1 > *n V n e N, be aLet r =statement that is either true or false. Use this preamble to answer questions 13,14,15 and 16.13. To prove that P, is true for all n EN, which of the following do you consider as thebase step?A. r2 < *1В. З> 1C. k+1 > *k -D. k+2 > *k+1E. None of the above.

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Description: Let an be a sequence of real numbers such that an+1 2 an, Vn 21 and an + a asn + 00. Use this preamble to answer questions 11 and 1211. Which of these statements best describe(s) a, as a Cauchy sequence?I. for any given e > 0, 3 n(e) E R Ə | an – a

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Let a, be a sequence of real numbers such that an+1 2 an, Vn 21 and a, + a as n + o. Use this preamble to answer questions 11 and 12 11. Which of these statements best describe(s) a, as a Cauchy sequence? I. for any given e > 0,3 n(e) ER 3| a, - a |< e Vn 2 n(e). II. for any givene > 0,| an - am < e V m,n 2 N, for some NE R III. | an - am +0 as m, n 0 A. I and II only B. I and III only C. II only D. III only E. None of the above choices A, B,C or D

Let a, be a sequence of real numbers such that an+1 2 an, Vn 21 and a, + a as n + o. Use this preamble to answer questions 11 and 12 11. Which of these statements best describe(s) a, as a Cauchy sequence? I. for any given e > 0,3 n(e) ER 3| a, - a |< e Vn 2 n(e). II. for any givene > 0,| an - am < e V m,n 2 N, for some NE R III. | an - am +0 as m, n 0 A. I and II only B. I and III only C. II only D. III only E. None of the above choices A, B,C or D

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 sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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