Let 0 < y 1 < x 1 and set a) Prove that 0 < y n

Question:

Let 0 < y₁ < x₁ and set

a) Prove that 0 < y_n < x_n for all n e N.

b) Prove that y_n is increasing and bounded above, and that x_n is decreasing and bounded below.

c) Prove that 0 < x_n+1 – y_n+1 < (x₁ – y₁)/2ⁿ for n ∊ N.

d) Prove that lim_n→∞ x_n = lim_n→∞ y_n. (This common value is called the arithmetic–geometric mean of x₁ and y₁.)

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An Introduction to Analysis

ISBN: 978-0132296380

4th edition

Authors: William R. Wade

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Question Posted: Mar 29, 2016 03:55 AM

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Let 0 < y 1 < x 1 and set a) Prove that 0 < y n

Question:

xn + yn 2and nnEN.

Step by Step Answer:

a This follows immediately from Exercise 126 b By a x n1 x n y n 2 2x n 2 x n Thus y n1 x n1 x 1 ...View the full answer

An Introduction to Analysis

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