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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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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