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Problem 11.2.22. The nth Term Test Show that if an converges then lim an = 0. %3D n→∞

Problem 11.2.22. The nth Term Test Show that if an converges then lim an = 0. %3D n→∞

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 11.2.22.

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 11.2.22. The th Term Test Show that if an converges then lim an = 0. = 0. in=1
Transcribed Image Text:Problem 11.2.22. The th Term Test Show that if an converges then lim an = 0. = 0. in=1
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