java upc-a we mean in Java

Maker UPC-A Supplement 2 in Java we mean

we mean
UPC-A Supplement 2 Maker In Java
Using Barcode encoder for Java Control to generate, create UPC A image in Java applications.
UPC-A Supplement 5 Recognizer In Java
Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications.
N
Draw Bar Code In Java
Using Barcode generation for Java Control to generate, create barcode image in Java applications.
Decoding Barcode In Java
Using Barcode reader for Java Control to read, scan read, scan image in Java applications.
lim S N
Universal Product Code Version A Creation In Visual C#
Using Barcode drawer for VS .NET Control to generate, create UPCA image in VS .NET applications.
Draw GTIN - 12 In VS .NET
Using Barcode generator for ASP.NET Control to generate, create UPCA image in ASP.NET applications.
provided that this limit exists
Make UPCA In Visual Studio .NET
Using Barcode creation for VS .NET Control to generate, create UPC Symbol image in .NET framework applications.
Drawing UPC-A Supplement 5 In VB.NET
Using Barcode generation for .NET framework Control to generate, create UPC-A image in Visual Studio .NET applications.
Discrete Mathematics Demystified
EAN-13 Creator In Java
Using Barcode maker for Java Control to generate, create EAN 13 image in Java applications.
Linear Printer In Java
Using Barcode maker for Java Control to generate, create Linear 1D Barcode image in Java applications.
132 Some Examples
Code 128C Encoder In Java
Using Barcode maker for Java Control to generate, create Code128 image in Java applications.
Code-39 Creator In Java
Using Barcode maker for Java Control to generate, create Code39 image in Java applications.
EXAMPLE 133 Does the series
UPC-E Printer In Java
Using Barcode drawer for Java Control to generate, create UPC E image in Java applications.
Creating ECC200 In .NET
Using Barcode encoder for .NET framework Control to generate, create Data Matrix image in .NET applications.
j=1
Decoding Bar Code In Java
Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications.
Painting Bar Code In None
Using Barcode drawer for Online Control to generate, create barcode image in Online applications.
2 j converge If so, to what limit
UPCA Decoder In Java
Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications.
EAN 13 Encoder In Objective-C
Using Barcode drawer for iPhone Control to generate, create EAN-13 Supplement 5 image in iPhone applications.
Solution: We have
Decoding Data Matrix 2d Barcode In Visual C#.NET
Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.
Recognizing GS1 128 In C#.NET
Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications.
SN =
2 j = 2 1 + 2 2 + 2 3 + + 2 N
= 2 1 + (2 1 2 2 ) + (2 2 2 3 ) + (2 3 2 4 ) + + (2 (N 1) 2 N ) (131)
Notice that S N is a nite sum so that it is correct to use the associative law of addition We conclude that S N = 2 1 + 2 1 + ( 2 2 + 2 2 ) + ( 2 3 + 2 3 ) + + ( 2 (N 1) + 2 (N 1) ) 2 N = 1 2 N We may now pass to the limit:
N
lim S N = lim (1 2 N ) = 1 lim 2 N = 1 0 = 1
N N
By de nition, we say that the series
j=1
2 j
converges to 1 Refer to Fig 131
Figure 131 Convergence of the series
2 j
Series
Insight: Do not worry about how one thinks of the trick that was used in Example 133 The purpose of this chapter is to teach you all the techniques that you will need in order to study series We will learn them gradually, beginning in the next section For now, just concentrate on learning what it means for a series to converge or diverge EXAMPLE 134 Discuss convergence for the series
j=1
( 1) j+1 = 1 1 + 1 1 + 1
Solution: If N is odd then SN = 1 1 + 1 1 + + 1 = (1 1) + (1 1) + + (1 1) + 1 =1 However, if N is even then SN = 1 1 + 1 1 + 1 = (1 1) + (1 1) + + (1 1) =0 Therefore the sequence {S N } =1 of partial sums is just the sequence N 1, 0, 1, 0, 1, 0, which does not converge We conclude that the series itself does not converge Insight: Example 134 explains away the apparent contradiction that we encountered in Example 131 The lesson is that we should not attempt to perform arithmetic operations on series that do not converge WARNING: It is easy to become confused at this point about the difference between a sequence and a series Remember that a sequence is a list of numbers while a series is a sum of numbers However, we study a series by looking at its sequence of partial sums
Discrete Mathematics Demystified
Sometimes there are tricks involved in seeing that a series converges: EXAMPLE 135 Discuss convergence for the series
j=1
1 j ( j + 1)
Solution: If you use a calculator to compute some partial sums (up to S20 , for instance) then you will probably be convinced that the series converges Here is a way to get this conclusion mathematically: We write SN = = 1 1 1 1 + + + + 1 2 2 3 3 4 N (N + 1) 1 1 1 2 + 1 1 2 3 + 1 1 3 4 + + 1 1 N N +1
Almost everything cancels out and we have SN = 1 1 N +1
Clearly lim N S N = 1 Therefore the series
j=1
1 j ( j + 1)
converges to 1 Insight: Sometimes it is convenient to begin a series at an index other than j = 1 An example is
j=3
j ln( j 1)
Obviously it would not do to begin this series at j = 1 because ln 0 is unde ned We cannot begin at j = 2 because ln 1 = 0 So we begin at j = 3 Of course this
series is equal to
Series
3 4 + + ln 2 ln 3 But notice this: the partial sum S1 is understood to be 0 because summing from 3 to 1 makes no sense The partial sum S2 is zero for a similar reason The rst nontrivial partial sum is S3 = Indeed, for N 3 we have SN = 3 4 N + + + ln 2 ln 3 ln(N 1) 3 ln 2
Copyright © OnBarcode.com . All rights reserved.