Let a n = (1 + 1/n)n. (a) Show that if 0 a < b, then

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Let an = (1 + 1/n)n.

(a) Show that if 0 ≤ a < b, then

(b) Deduce that bn[(n + 1)a − nb], an+1.

(c) Use a = 1 + 1/(n + 1) and b = 1 + 1/n in part (b) to show that {an} is increasing.

(d) Use a = 1 and b = 1 + 1/(2n) in part (b) to show that a2n < 4.

(e) Use parts (c) and (d) to show that an < 4 for all n.

(f) Use Theorem 12 to show that limn→∞(1 + 1/n)n exists.

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Calculus Early Transcendentals

ISBN: 9781337613927

9th Edition

Authors: James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin

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