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Math Calculus Calculus - Standalone book

To prove that lim N → ∞ R N and lim N → ∞ L N exist and are equal if f(x) is increasing (the case of f(x) decreasing is similar). a) A graph to explain why L N ≤ R M for all N , M ≥ 1. b) The sequence { L N } is bounded, so it has a least upper bound L . By definition, L is the smallest number such that L N ≤ L for all N . Show that L ≤ R M for all M . c) Show that lim N → ∞ L N = L and lim N → ∞ R N = R .

To prove that lim N → ∞ R N and lim N → ∞ L N exist and are equal if f(x) is increasing (the case of f(x) decreasing is similar). a) A graph to explain why L N ≤ R M for all N , M ≥ 1. b) The sequence { L N } is bounded, so it has a least upper bound L . By definition, L is the smallest number such that L N ≤ L for all N . Show that L ≤ R M for all M . c) Show that lim N → ∞ L N = L and lim N → ∞ R N = R .

Solution Summary: The author explains how to prove that lim N R N and L N exist and are equal if f(x) is increasing (the case of decreasing is similar).

Calculus - Standalone book

Calculus - Standalone book

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Sequence and Series

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Chapter 5.1, Problem 87E

To determine

To prove that $\lim_{N \to \infty} R_{N}$ and $\lim_{N \to \infty} L_{N}$ exist and are equal if f(x) is increasing (the case of f(x) decreasing is similar).

a) A graph to explain why $L_{N} \leq R_{M}$ for all $N, M \geq 1.$

b) The sequence ${L_{N}}$ is bounded, so it has a least upper bound L. By definition, L is the smallest number such that $L_{N} \leq L$ for all N. Show that $L \leq R_{M}$ for all M.

c) Show that $\lim_{N \to \infty} L_{N} = L$ and $\lim_{N \to \infty} R_{N} = R .$

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Chapter 5.1, Problem 86E

Chapter 5.1, Problem 88E

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Chapter 5 Solutions

Calculus - Standalone book

Ch. 5.1 - Prob. 1PQ Ch. 5.1 - Prob. 2PQ Ch. 5.1 - Prob. 3PQ Ch. 5.1 - Prob. 4PQ Ch. 5.1 - Prob. 5PQ Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E

Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - Prob. 49E Ch. 5.1 - Prob. 50E Ch. 5.1 - Prob. 51E Ch. 5.1 - Prob. 52E Ch. 5.1 - Prob. 53E Ch. 5.1 - Prob. 54E Ch. 5.1 - Prob. 55E Ch. 5.1 - 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