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Theorem 11.2.20. Cauchy Criterion. The series Eo ak converges if and only if Ve > 0, N such that if m >n > N then |En+1 ak| < e. Problem 11.2.21. Prove the Cauchy criterion.

Theorem 11.2.20. Cauchy Criterion. The series Eo ak converges if and only if Ve > 0, N such that if m >n > N then |En+1 ak| < e. Problem 11.2.21. Prove the Cauchy criterion.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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Real math analysis, I need help with problem 11.2.21.

thank you so much.

Theorem 11.2.15. Cauchy sequences converge Suppose (sn) is a Cauchy sequence of real numbers. There exists a real number s such that limn→∞ Sn = s. Sketch of Proof. We know that (sn) is bounded, so by the Bolzano-Weierstrass Theorem, it has a convergent subsequence (Snk) converging to some real number s. We have sn – s = |Sn – Snp + Snk s| < |Sn – Sni|+|Snk 8|. If we choose n and ng large enough, we should be able to make each term arbitrarily small.
Transcribed Image Text:Theorem 11.2.15. Cauchy sequences converge Suppose (sn) is a Cauchy sequence of real numbers. There exists a real number s such that limn→∞ Sn = s. Sketch of Proof. We know that (sn) is bounded, so by the Bolzano-Weierstrass Theorem, it has a convergent subsequence (Snk) converging to some real number s. We have sn – s = |Sn – Snp + Snk s| < |Sn – Sni|+|Snk 8|. If we choose n and ng large enough, we should be able to make each term arbitrarily small.
Problem 11.2.19. Since the convergence of Cauchy sequences can be taken as the completeness axiom for the real number system, it does not hold for the rational number system. Give an example of a Cauchy sequence of rational numbers which does not converge to a rational number. If we apply the above ideas to series we obtain the following important result, which will provide the basis for our investigation of power series. Theorem 11.2.20. Cauchy Criterion. The series -o ak converges if and only if Ve > 0, N such that if m > n > N then |En+1. ak < ɛ. Problem 11.2.21. Prove the Cauchy criterion.
Transcribed Image Text:Problem 11.2.19. Since the convergence of Cauchy sequences can be taken as the completeness axiom for the real number system, it does not hold for the rational number system. Give an example of a Cauchy sequence of rational numbers which does not converge to a rational number. If we apply the above ideas to series we obtain the following important result, which will provide the basis for our investigation of power series. Theorem 11.2.20. Cauchy Criterion. The series -o ak converges if and only if Ve > 0, N such that if m > n > N then |En+1. ak < ɛ. Problem 11.2.21. Prove the Cauchy criterion.
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