Explanation Given: We need 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) Explain with a graph 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 = L . Calculation: a) Let f(x ) be positive and increasing, and let N and M be positive integer. From the figure below at the left, we see that L N underestimates the area under the graph of y = f ( x ) , while from the figure below at the right, we see that R M overestimates the area under the graph. Thus, for all N , M ≥ 1 , L N ≤ R M b) Because the sequence { L N } is bounded above by R M for any M , each R M is an upper bound for the sequence. Furthermore, the sequence { L N } must have a least upper bound, call it L . By definition, the least upper bound must be no greater than any other upper bound, consequently, L ≤ R M for all M. c) Since L N ≤ L ≤ R M , R N − L ≤ R N − L N , so | R N − L | ≤ | R N − L N | . From this, lim N → ∞ | R N − L | ≤ lim N → ∞ | R N − L N | ...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 5.1, Problem 91E

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 90E

Chapter 5.1, Problem 92E

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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).

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