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barcode in ssrs report POWER SERIES 11.39. Prove that both the power series in .NET framework
POWER SERIES 11.39. Prove that both the power series Reading QR Code JIS X 0510 In Visual Studio .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET framework applications. QR Code Generator In .NET Using Barcode drawer for .NET framework Control to generate, create QR Code 2d barcode image in .NET applications. 1 X n 0
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Make DataBar In VS .NET Using Barcode generator for .NET Control to generate, create DataBar image in .NET applications. Print Identcode In VS .NET Using Barcode creator for VS .NET Control to generate, create Identcode image in .NET framework applications. Let R > 0 be the radius of convergence of an xn . Let 0 < jx0 j < R. Then, as in Problem 11.33, we can 1 choose N as that jan j < for n > N. jx0 jn Thus, the terms of the series jnan xn 1 j njan jjxjn 1 can for n > N be made less than corresponding jxjn 1 terms of the series n , which converges, by the ratio test, for jxj < jx0 j < R. jx0 jn Hence, nan xn 1 converges absolutely for all points x0 (no matter how close jx0 j is to R). If, however, jxj > R, lim an xn 6 0 and thus lim nan xn 1 6 0, so that nan xn 1 does not converge. Making 1D Barcode In Visual Studio .NET Using Barcode encoder for ASP.NET Control to generate, create Linear Barcode image in ASP.NET applications. Encode UPC  13 In Java Using Barcode encoder for BIRT Control to generate, create EAN13 image in BIRT reports applications. n!1 n!1 USS128 Generator In VB.NET Using Barcode creation for VS .NET Control to generate, create EAN 128 image in VS .NET applications. Making ECC200 In None Using Barcode maker for Font Control to generate, create Data Matrix image in Font applications. INFINITE SERIES
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UPC  13 Generator In None Using Barcode creator for Software Control to generate, create EAN / UCC  13 image in Software applications. Data Matrix 2d Barcode Drawer In Java Using Barcode maker for Java Control to generate, create Data Matrix image in Java applications. Thus, R is the radius of convergence of nan xn 1 . Note that the series of derivatives may or may not converge for values of x such that jxj R. 11.40. Illustrate Problem 11.39 by using the series
1 X xn . n2 3n n 1
u xn 1 n2 3n n2 jxj lim n 1 lim jxj lim u n!1 n 1 2 3n 1 n n!1 n!1 3 x 3 n 1 2 n so that the series converges for jxj < 3. convergence is 3 @ x @ 3. The series of derivatives is At x 3 the series also converges, so that the interval of 1 1 X nxn 1 X xn 1 2 n n 3n n 3 n 1 n 1
By Problem 11.25(a) this has the interval of convergence 3 @ x < 3. The two series have the same radius of convergence, i.e., R 3, although they do not have the same interval of convergence. Note that the result of Problem 11.39 can also be proved by the ratio test if this test is applicable. The proof given there, however, applies even when the test is not applicable, as in the series of Problem 11.22. 11.41. Prove that in any interval within its interval of convergence a power series a represents a continuous function, say, f x , b can be integrated term by term to yield the integral of f x , c can be di erentiated term by term to yield the derivative of f x . We consider the power series an xn , although analogous results hold for an x a n . (a) This follows from Problem 11.33 and 11.34, and the fact that each term an xn of the series is continuous. (b) This follows from Problems 11.33 and 11.35, and the fact that each term an xn of the series is continuous and thus integrable. (c) From Problem 11.39, the series of derivatives of a power series always converges within the interval of convergence of the original power series and therefore is uniformly convergent within this interval. Thus, the required result follows from Problems 11.33 and 11.36. If a power series converges at one (or both) end points of the interval of convergence, it is possible to establish (a) and (b) to include the end point (or end points). See Problem 11.42. 11.42. Prove Abel s theroem that if a power series converges at an end point of its interval of convergence, then the interval of uniform convergence includes this end point. For simplicity in the proof, we assume the power series to be
1 X k 0
ak xk with the end point of its interval Then we must show that the
of convergence at x 1, so that the series surely converges for 0 @ x @ 1. series converges uniformly in this interval. Let Rn x an xn an 1 xn 1 an 2 xn 2 ; Rn an an 1 an 2
To prove the required result we must show that given any > 0, we can nd N such that jRn x j < for all n > N, where N is independent of the particular x in 0 @ x @ 1. CHAP. 11]

