This preview shows page 1. Sign up to view the full content.
Unformatted text preview: n n + 1 = 1 n + 1 1 n = 1 + 1 n + 1 1 + 1 n = p 1 + (1 / ( n + 1)) p 1 + (1 /n ) . Since this sequence is monotonically decreasing and bounded below, it must converge to its greatest lower bound, which is 1. For this latter step, you could also cite Theorem 3.2.10 of the text. 2. Let b R satisfy 0 < b < 1. Show that lim nb n = 0. We write b = 1 1+ a . Then a = 1 b1 > 0 since b < 1. Then b n = 1 (1 + a ) n 1 (1 / 2) n ( n1) a 2 by the binomial theorem. And, nb n 2 ( n1) a 2 . The right side clearly tends to 0 as n , and thus, by the squeeze lemma, nb n 0. 1...
View
Full
Document
This note was uploaded on 07/15/2010 for the course MATH 521 taught by Professor Metcalfe during the Spring '10 term at University of North Carolina Wilmington.
 Spring '10
 MetCalfe
 Calculus

Click to edit the document details