Irrationality of e Prove that e is an irrational number using the following argument by

Chapter 10, Problem 93

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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! e1 is a whole number. (b) Use the power series for f (x) = ex at x = 1 to show that there is an integer B such that M! e1 equals B + (1) M+1 1 M + 1 1 (M + 1)(M + 2) + (c) Use your knowledge of alternating series with decreasing terms to conclude that 0 < |M! e1 B| < 1 and observe that this contradicts (a). Hence, e is not equal to M/N.