p 3un , u1 1. (a) Prove that lim un in VS .NET

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p 3un , u1 1. (a) Prove that lim un
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p p (a) The terms of the sequence are u1 1, u2 3u1 n 1 1=2 , u3 3u2 31=2 1=4 ; . . . . 3 1=2 1=4 1=2 as can be proved by mathematical induction The nth term is given by un 3 ( 1). Clearly, un 1 A un . Then the sequence is monotone increasing. By Problem 1.14, 1, un @ 31 3, i.e. un is bounded above. Hence, un is bounded (since a lower bound is zero). Thus, a limit exists, since the sequence is bounded and monotonic increasing.
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p p (b) Let x required limit. Since lim un 1 lim 3un , we have x 3x and x 3. n!1 n!1 possibility, x 0, is excluded since un A 1: Another method: lim 31=2 1=4 1=2
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lim 31 1=2 3 n!1
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lim 1 1=2
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2.17. Verify the validity of the entries in the following table.
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Monotonic Increasing No No No Yes No Monotonic Decreasing Yes No No No No Limit Exists No No Yes (0) Yes (2) 3 No
Sequence 2; 1:9; 1:8; 1:7; . . . ; 2 n 1 =10 . . . 1; 1; 1; 1; . . . ; 1
Bounded No Yes 1 ; . . . Yes Yes No
;...
n 1 1 1 1 1 = n 2 ; 3 ; 4 ; 5 ; . . . ; 1
:6; :66; :666; . . . ; 2 1 1=10n ; . . . 3 1; 2; 3; 4; 5; . . . ; 1 n n; . . .
2.18. Prove that the sequence with the nth term un 1
1 is monotonic, increasing, and bounded, n and thus a limit exists. The limit is denoted by the symbol e.   1 n Note: lim 1 e, where e 2:71828 . . . was introduced in the eighteenth century by n!1 n Leonhart Euler as the base for a system of logarithms in order to simplify certain di erentiation and integration formulas.
By the binomial theorem, if n is a positive integer (see Problem 1.95, 1), 1 x n 1 nx n n 1 2 n n 1 n 2 3 n n 1 n n 1 n x x x 2! 3! n!
Letting x 1=n,   1 n 1 n n 1 1 n n 1 n n 1 1 un 1 1 n n n 2! n! nn n2      1 1 1 1 2 1 1 1 1 1 2! n 3! n n      1 1 2 n 1 1 1 1 n! n n n Since each term beyond the rst two terms in the last expression is an increasing function of n, it follows that the sequence un is a monotonic increasing sequence. It is also clear that   1 n 1 1 1 1 1 1 1 < 1 1 < 1 1 2 n 1 < 3 n 2! 3! n! 2 2 2 by Problem 1.14, 1. Thus, un is bounded and monotonic increasing, and so has a limit which we denote by e. The value of e 2:71828 . . . .
  1 x 2.19. Prove that lim 1 e, where x ! 1 in any manner whatsoever (i.e., not necessarily along x!1 x the positive integers, as in Problem 2.18).
If n largest integer @ x, then n @ x @ n 1 and Since n     1 1 x 1 n 1 @ 1 @ 1 . n 1 x n  n  n 1 ,  1 1 1 e lim 1 1 lim 1 n!1 n!1 n 1 n 1 n 1  1
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