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93. Irrationality of e Prove that e is an irrational number using the following argument by contradiction. Suppose that e = M/N, where M, N are nonzero integers. (a) Show that M!e- is a whole number. (b) Use the power series for f(x) = e at x = -1 to show that there is an integer B such that M!e- equals B+ (-1)M+1 (M + 1)(M + 2) м+1 (c) Use your knowledge of alternating series with decreasing terms to conclude that 0 < |M!e- - B| < 1 and observe that this contradicts (a). Hence, e is not equal to M/N.

93. Irrationality of e Prove that e is an irrational number using the following argument by contradiction. Suppose that e = M/N, where M, N are nonzero integers. (a) Show that M!e- is a whole number. (b) Use the power series for f(x) = e at x = -1 to show that there is an integer B such that M!e- equals B+ (-1)M+1 (M + 1)(M + 2) м+1 (c) Use your knowledge of alternating series with decreasing terms to conclude that 0 < |M!e- - B| < 1 and observe that this contradicts (a). Hence, e is not equal to M/N.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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93. Irrationality of e Prove that e is an irrational number using the following argument by contradiction. Suppose that e = M/N, where M, N are nonzero integers. (a) Show that M!e- is a whole number. (b) Use the power series for f(x) = e at x = -1 to show that there is an integer B such that M!e- equals B+ (-1)M+1 (M + 1)(M + 2) м+1 (c) Use your knowledge of alternating series with decreasing terms to conclude that 0 < |M!e- - B| < 1 and observe that this contradicts (a). Hence, e is not equal to M/N.
Transcribed Image Text:93. Irrationality of e Prove that e is an irrational number using the following argument by contradiction. Suppose that e = M/N, where M, N are nonzero integers. (a) Show that M!e- is a whole number. (b) Use the power series for f(x) = e at x = -1 to show that there is an integer B such that M!e- equals B+ (-1)M+1 (M + 1)(M + 2) м+1 (c) Use your knowledge of alternating series with decreasing terms to conclude that 0 < |M!e- - B| < 1 and observe that this contradicts (a). Hence, e is not equal to M/N.
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