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X n 1 in .NET framework
1 X n 1 QR Code 2d Barcode Scanner In .NET Framework Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Print QR Code JIS X 0510 In .NET Framework Using Barcode printer for Visual Studio .NET Control to generate, create QRCode image in Visual Studio .NET applications. un x u1 x u2 x u3 x
Denso QR Bar Code Decoder In VS .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications. Barcode Maker In Visual Studio .NET Using Barcode encoder for .NET framework Control to generate, create barcode image in Visual Studio .NET applications. is said to be convergent in a; b if the sequence of partial sums fSn x g, n 1; 2; 3; . . . ; where Sn x u1 x u2 x un x , is convergent in a; b . In such case we write lim Sn x S x n!1 and call S x the sum of the series. It follows that un x converges to S x in a; b if for each > 0 and each x in a; b we can nd N > 0 such that jSn x S x j < for all n > N. If N depends only on and not on x, the series is called uniformly convergent in a; b . Since S x Sn x Rn x , the remainder after n terms, we can equivalently say that un x is uniformly convergent in a; b if for each > 0 we can nd N depending on but not on x such that jRn x j < for all n > N and all x in a; b . These de nitions can be modi ed to include other intervals besides a @ x @ b, such as a < x < b, and so on. The domain of convergence (absolute or uniform) of a series is the set of values of x for which the series of functions converges (absolutely or uniformly). Bar Code Decoder In Visual Studio .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET framework applications. QR Code 2d Barcode Drawer In C#.NET Using Barcode printer for Visual Studio .NET Control to generate, create QR image in .NET applications. EXAMPLE 1. Suppose un xn =n and 1 @ x @ 1. Now think of the constant function F x 0 on this interval. 2 For any > 0 and any x in the interval, there is N such that for all n > Njun F x j < , i.e., jxn =nj < . Since the limit does not depend on x, the sequence is uniformly convergent. Printing QR Code ISO/IEC18004 In .NET Using Barcode generation for ASP.NET Control to generate, create QR Code image in ASP.NET applications. QR Code JIS X 0510 Generation In Visual Basic .NET Using Barcode drawer for Visual Studio .NET Control to generate, create QR Code image in .NET framework applications. INFINITE SERIES
Creating DataMatrix In .NET Framework Using Barcode drawer for .NET framework Control to generate, create Data Matrix 2d barcode image in VS .NET applications. Paint 1D In VS .NET Using Barcode generation for .NET framework Control to generate, create Linear Barcode image in VS .NET applications. [CHAP. 11
Matrix Barcode Creation In .NET Framework Using Barcode drawer for .NET framework Control to generate, create 2D Barcode image in VS .NET applications. Draw Code 9/3 In Visual Studio .NET Using Barcode maker for Visual Studio .NET Control to generate, create ANSI/AIM Code 93 image in VS .NET applications. EXAMPLE 2. If un xn and 0 @ x @ 1, the sequence is not uniformly convergent because (think of the function F x 0, 0 @ x < 1, F 1 1 jxn 0j < when xn < ; thus n ln x < ln : Draw Code 128 In None Using Barcode generation for Office Excel Control to generate, create USS Code 128 image in Excel applications. EAN13 Supplement 5 Generator In .NET Using Barcode generator for Reporting Service Control to generate, create GTIN  13 image in Reporting Service applications. On the interval 0 @ x < 1, and for 0 < < 1, both members ln : Since of the inequality are negative, therefore, n > ln x ln ln 1 ln ln = , it follows that we must choose N ln x ln 1 nn x ln 1=x such that n>N> ln 1= ln 1=x Code 39 Full ASCII Decoder In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Encoding Code 128 In ObjectiveC Using Barcode drawer for iPhone Control to generate, create Code 128 image in iPhone applications. 1 From this expression we see that ! 0 then ln ! 1 and 1 also as x ! 1 from the left ln ! 0 from the right; thus, in either x case, N must increase without bound. This dependency on both and x demonstrations that the sequence is not uniformly convergent. For a pictorial view of this example, see Fig. 111. Generate Bar Code In ObjectiveC Using Barcode creator for iPhone Control to generate, create barcode image in iPhone applications. EAN128 Creator In ObjectiveC Using Barcode generation for iPhone Control to generate, create EAN / UCC  13 image in iPhone applications. Fig. 111 Generating Barcode In Java Using Barcode creator for Android Control to generate, create barcode image in Android applications. GTIN  128 Generation In Java Using Barcode drawer for Java Control to generate, create EAN / UCC  14 image in Java applications. SPECIAL TESTS FOR UNIFORM CONVERGENCE OF SERIES 1. Weierstrass M test. If sequence of positive constants M1 ; M2 ; M3 ; . . . can be found such that in some interval (a) jun x j @ Mn n 1; 2; 3; . . . (b) 1 X cos nx n 1
Mn converges
then un x is uniformly and absolutely convergent in the interval.
EXAMPLE. converges. n2
X1 cos nx 1 is uniformly and absolutely convergent in 0; 2 since 2 @ 2 and n n n2
This test supplies a su cient but not a necessary condition for uniform convergence, i.e., a series may be uniformly convergent even when the test cannot be made to apply. One may be led because of this test to believe that uniformly convergent series must be absolutely convergent, and conversely. However, the two properties are independent, i.e., a series can be uniformly convergent without being absolutely convergent, and conversely. See Problems 11.30, 11.127. Dirichlet s test. Suppose that (a) (b) the sequence fan g is a monotonic decreasing sequence of positive constants having limit zero, there exists a constant P such that for a @ x @ b ju1 x u2 x un x j < P Then the series a1 u1 x a2 u2 x is uniformly convergent in a @ x @ b.

