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Suppose the sequence {an) satisfies the inequality lan+1-an an | < |ªn-an-1|, n ≥ 1.

Suppose the sequence {an) satisfies the inequality lan+1-an an | < |ªn-an-1|, n ≥ 1.

Advanced Engineering Mathematics
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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Exercise 3: Suppose the sequence {ano satisfies the inequality |an+1 - an| <= |an - an-1|, n ≥ 1. • Using induction, show that for n ≥ 1, lan+1 - an < 2n 1 = 2, < 2n . Remembering that Σ-2/3 j=0 |a₁-aol. and the power of telescoping series so that k-1 lan+kan ≤lan+s+1 - an+sl j=0 prove that {an) is a Cauchy sequence.
Transcribed Image Text:Exercise 3: Suppose the sequence {ano satisfies the inequality |an+1 - an| <= |an - an-1|, n ≥ 1. • Using induction, show that for n ≥ 1, lan+1 - an < 2n 1 = 2, < 2n . Remembering that Σ-2/3 j=0 |a₁-aol. and the power of telescoping series so that k-1 lan+kan ≤lan+s+1 - an+sl j=0 prove that {an) is a Cauchy sequence.
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