vb.net print barcode zebra ESSENTIAL MATHEMATICS in Java Generation DataMatrix in Java ESSENTIAL MATHEMATICS

ESSENTIAL MATHEMATICS
Recognize ECC200 In Java
Using Barcode Control SDK for Java Control to generate, create, read, scan barcode image in Java applications.
Paint Data Matrix ECC200 In Java
Using Barcode creation for Java Control to generate, create ECC200 image in Java applications.
GEOMETRIC SERIES A series is a sequence of possibly infinitely many terms whose sum is to be determined. A geometric series is a series in which each term is the same multiple of its predecessor. For example, 20 + 60 + 180 + 540 + 1620 + 4860 + is a geometric series because each term is 3 times the size of its predecessor. The multiplier 3 is called the common ratio of the series. Theorem A.5 Sum of a Finite Geometric Series If r 1, then a 1 rn = --------------------1 r Here, a is the first term in the series, r is the common ratio, and n is the number of terms in the series. a + ar + ar + ar +
Using Barcode scanner for Java Control to read, scan read, scan image in Java applications.
Bar Code Creation In Java
Using Barcode drawer for Java Control to generate, create barcode image in Java applications.
+ ar
Barcode Recognizer In Java
Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications.
Data Matrix Encoder In Visual C#.NET
Using Barcode printer for Visual Studio .NET Control to generate, create Data Matrix image in .NET framework applications.
EXAMPLE A.6 Finite Geometric Series
Paint DataMatrix In .NET Framework
Using Barcode drawer for ASP.NET Control to generate, create ECC200 image in ASP.NET applications.
Make ECC200 In VS .NET
Using Barcode maker for .NET Control to generate, create Data Matrix ECC200 image in .NET applications.
For the sum 20 + 60 + 180 + 540 + 1620 + 4860, the three parameters are a = 20, r = 3, and n = 6. So the sum is a 1 r 20 1 3 20 1 729 20 729 -------------------- = ------------------------ = ---------------------------- = ---------------------- = 7280 1 r 1 3 2 2
ECC200 Generation In Visual Basic .NET
Using Barcode maker for VS .NET Control to generate, create Data Matrix ECC200 image in Visual Studio .NET applications.
EAN-13 Drawer In Java
Using Barcode maker for Java Control to generate, create UPC - 13 image in Java applications.
Theorem A.6 Sum of an Infinite Geometric Series If 1 < r < 1, then a 2 3 a + ar + ar + ar + = ---------1 r EXAMPLE A.7 Infinite Geometric Series
1D Encoder In Java
Using Barcode creator for Java Control to generate, create Linear image in Java applications.
UPCA Encoder In Java
Using Barcode maker for Java Control to generate, create UPCA image in Java applications.
For the sum 0.42 + 0.0042 + 0.000042 + 0.00000042 + 0.0000000042 + a = 0.42 and r = 0.01. So the infinite sum is a0.42 ---------- = ------------------ = 0.42 = 42 = 14 ----------------1 r 1 0.01 0.99 99 33 , the three parameters are
Generate EAN 8 In Java
Using Barcode encoder for Java Control to generate, create EAN-8 Supplement 2 Add-On image in Java applications.
Draw UCC - 12 In Java
Using Barcode drawer for BIRT reports Control to generate, create GS1 - 12 image in BIRT reports applications.
Note that 14/33 = 0.4242424242 . This repeating decimal is obviously the same as the infinite sum 0.42 + 0.0042 + 0.000042 + 0.00000042 + 0.0000000042 + .
Drawing ECC200 In None
Using Barcode encoder for Font Control to generate, create Data Matrix ECC200 image in Font applications.
Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications.
OTHER SUMMATION FORMULAS Theorem A.7 Sum of the First n Positive Integers n n+1 + n = ------------------2 Note that the parameter n equals the number of terms in the sum. 1+2+3+ EXAMPLE A.8 Summing Positive Integers
Bar Code Generator In Visual Studio .NET
Using Barcode maker for Reporting Service Control to generate, create bar code image in Reporting Service applications.
Print Universal Product Code Version A In Objective-C
Using Barcode maker for iPhone Control to generate, create GTIN - 12 image in iPhone applications.
The sum of the first 10 integers is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10(10+1)/2 = 55.
DataMatrix Scanner In C#.NET
Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications.
Scanning Bar Code In VS .NET
Using Barcode Control SDK for ASP.NET Control to generate, create, read, scan barcode image in ASP.NET applications.
ESSENTIAL MATHEMATICS
[APPENDIX
Theorem A.8 Sum of the First n Squares n n + 1 2n + 1 2 + n = ---------------------------------------6 The expression on the right appears to be a fraction. But it will always turn out to be an integer because it equals a sum of integers. 1 +2 +3 +
2 2 2
EXAMPLE A.9 Summing Squares
The sum of the first six squares is 12 + 22 + 32 + 42 + 52 + 62 = 6(7)(13)/6 = 546/6 = 91.
HARMONIC NUMBERS The harmonic series is the series of reciprocals:
1 -- = 1 + 1 + 1 + 1 + 1 + -- -- -- -- - - 2 3 4 5 k=1k
1 2 3 4 5 6 7 8 9 10
1.000000 1.500000 1.833333 2.083333 2.283333 2.450000 2.592857 2.717857 2.828968 2.928968
It is not hard to see that this series diverges. That is, its partial sums increase without bound. The partial sums of the harmonic series are called the harmonic numbers and are denoted by Hn :
1 -- = 1 + 1 + 1 + 1 + 1 + -- -- -- -- - - Hn = 2 3 4 5 k=1k
1 + -n
The first three harmonic numbers are
H1 = 1 -- = 1 k = 1k 1 -- = 1 + 1 = 3 --2 2 k = 1k 1 -- = 1 + 1 + 1 = 5 -- -- -2 3 6 k = 1k
3 2 1
Table A.1 Harmonic numbers
H2 =
H3 =
Although the harmonic numbers increase without bound, it is not obvious how fast they increase. Table A.1 suggests that they increase very slowly. The fact is that the harmonic numbers increase logarithmin n! cally: H n = (lgn). This means that they increase at about the 0 1 same rate as logarithmic numbers. More precisely, it means that 1 1 both ratios H n/ lgn and lgn /H n are bounded.
2 3 4 5 6 7 8 9 2 6 24 120 720 5040 40,320 362,880
STIRLING S FORMULA The factorial numbers frequently appear in the analysis of algorithms. They are defined by:
n! =
k = 1 2 3 4