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2) The purpose of this problem is to give you some practice with the most common application of The Big Theorem in this course. First, a definition. Let {an} and {bn} be two sequences. We say that {bn} eventually dominates {an} if there is an N € N such that a,n < bn for all n> N. (a) Let {an} and {bn} be sequences. Assume that a, << bn. Prove that {b„} eventually dominates {a„}. (b) Prove that {n0.1} eventually dominates {In n}. (c) Let {an}, {bn}, {Cn}, and {dn} be sequences. Prove that If an << bn and en << dn, then a„Cn << bndn (d) Prove that for all a >0 and all integers k > 1, {nª} eventually dominates {(In n)*}.

2) The purpose of this problem is to give you some practice with the most common application of The Big Theorem in this course. First, a definition. Let {an} and {bn} be two sequences. We say that {bn} eventually dominates {an} if there is an N € N such that a,n < bn for all n> N. (a) Let {an} and {bn} be sequences. Assume that a, << bn. Prove that {b„} eventually dominates {a„}. (b) Prove that {n0.1} eventually dominates {In n}. (c) Let {an}, {bn}, {Cn}, and {dn} be sequences. Prove that If an << bn and en << dn, then a„Cn << bndn (d) Prove that for all a >0 and all integers k > 1, {nª} eventually dominates {(In n)*}.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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(2) The purpose of this problem is to give you some practice with the most common application of The Big Theorem in this course. First, a definition. Let {an} and {bn} be two sequences. We say that {bn} eventually dominates {an} if there is an N €N such that an < bn for all n > N. (a) Let {an} and {b„} be sequences. Assume that a,n << bn. Prove that {b,} eventually dominates {an}. (b) Prove that {nº.1} eventually dominates {In n}. (c) Let {an}, {bn}, {Cn}, and {dn} be sequences. Prove that If an << bn and en << dn (d) Prove that for all a > 0 and all integers k > 1, {n°} eventually dominates {(In n)*}. anCn << bndn
Transcribed Image Text:(2) The purpose of this problem is to give you some practice with the most common application of The Big Theorem in this course. First, a definition. Let {an} and {bn} be two sequences. We say that {bn} eventually dominates {an} if there is an N €N such that an < bn for all n > N. (a) Let {an} and {b„} be sequences. Assume that a,n << bn. Prove that {b,} eventually dominates {an}. (b) Prove that {nº.1} eventually dominates {In n}. (c) Let {an}, {bn}, {Cn}, and {dn} be sequences. Prove that If an << bn and en << dn (d) Prove that for all a > 0 and all integers k > 1, {n°} eventually dominates {(In n)*}. anCn << bndn
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