# p 3un , u1 1. (a) Prove that lim un in VS .NET Generation QR Code 2d barcode in VS .NET p 3un , u1 1. (a) Prove that lim un

p 3un , u1 1. (a) Prove that lim un
QR Code 2d Barcode Reader In VS .NET
Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications.
QR-Code Drawer In .NET
Using Barcode creation for Visual Studio .NET Control to generate, create QR Code image in Visual Studio .NET applications.
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.
QR Code JIS X 0510 Decoder In Visual Studio .NET
Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET framework applications.
Encode Bar Code In Visual Studio .NET
Using Barcode maker for Visual Studio .NET Control to generate, create barcode image in .NET applications.
CHAP. 2]
Recognizing Bar Code In .NET Framework
Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET applications.
Denso QR Bar Code Generation In C#.NET
Using Barcode creation for .NET framework Control to generate, create QR-Code image in Visual Studio .NET applications.
SEQUENCES
Make Denso QR Bar Code In Visual Studio .NET
Using Barcode creation for ASP.NET Control to generate, create QR image in ASP.NET applications.
QR-Code Generator In VB.NET
Using Barcode printer for Visual Studio .NET Control to generate, create QR Code 2d barcode image in .NET applications.
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
Make GS1 DataBar Expanded In Visual Studio .NET
Using Barcode generator for Visual Studio .NET Control to generate, create GS1 RSS image in VS .NET applications.
Encode GTIN - 13 In .NET Framework
Using Barcode generator for Visual Studio .NET Control to generate, create UPC - 13 image in VS .NET applications.
(The other
EAN / UCC - 14 Maker In .NET
Using Barcode generator for .NET framework Control to generate, create GTIN - 128 image in Visual Studio .NET applications.
Drawing ISSN - 10 In .NET Framework
Using Barcode creation for Visual Studio .NET Control to generate, create International Standard Serial Number image in .NET applications.
lim 31 1=2 3 n!1
EAN-13 Generator In Visual Studio .NET
Using Barcode encoder for Reporting Service Control to generate, create EAN-13 image in Reporting Service applications.
ANSI/AIM Code 128 Generation In None
Using Barcode drawer for Microsoft Word Control to generate, create Code 128A image in Word applications.
lim 1 1=2
Using Barcode Control SDK for BIRT Control to generate, create, read, scan barcode image in BIRT applications.
Making GS1 128 In C#
Using Barcode creator for Visual Studio .NET Control to generate, create GTIN - 128 image in VS .NET applications.
31 3
Barcode Recognizer In C#
Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications.
Generating EAN 128 In VB.NET
Using Barcode generation for .NET Control to generate, create EAN128 image in .NET framework applications.
2.17. Verify the validity of the entries in the following table.
Barcode Creation In Java
Using Barcode creation for Android Control to generate, create bar code image in Android applications.
EAN13 Generation In None
Using Barcode drawer for Software Control to generate, create EAN 13 image in Software applications.
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