1.Suppose that the sequence (an) is monotone. Prove that {an} converges if and only if {a} converges. Show that the result does not hold without the monotonicity assumption.
1.Suppose that the sequence (an) is monotone. Prove that {an} converges if and only if {a} converges. Show that the result does not hold without the monotonicity assumption.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 72E
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Transcribed Image Text:1.Suppose that the sequence {an} is monotone. Prove that {an} converges if and only if {a}
converges. Show that the result does not hold without the monotonicity assumption.
2. Suppose that the sequence {a} converges to a and that a < 1. Prove that the sequence
{(an)"} converges to zero. (Hint: You need to use the fact that a < 1. You will need to point
out in your proof how are using this.)
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