3.05.Convergence - Approximation Authors: Bill Davis,...

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Unformatted text preview: Approximation Authors: Bill Davis, Horacio Porta and Jerry Uhl 1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 3.05 Barriers to Convergence BASICS B.1) Barriers and complex numbers. B.1.a) Look at these attempts to approximate f @ x D = 1 4- 2x + x 2 by early parts of its expansion in powers of x: Clear @ f, approx, x D ; f @ x _ D = 1 4- 2 x + x 2 ; approx @ x _ D = Normal @ Series @ f @ x D , 8 x, 0, 8 <DD ; Plot @8 f @ x D , approx @ x D< , 8 x,- 2.2, 2.2 < , PlotStyle 88 Thickness @ 0.02 D , Blue < , 8 Thickness @ 0.01 D , Red << , PlotRange 8- 0.3, 0.8 < , AxesLabel 8 "x" , "" < , PlotLabel "Powers of x centered at x = 0" D- 2- 1 1 2 x- 0.2 0.2 0.4 0.6 0.8 Powers of x centered at x = The approximation breaks down on the left and the right. Try to recover by using more of the expansion: Clear @ approx D ; approx @ x _ D = Normal @ Series @ f @ x D , 8 x, 0, 14 <DD ; Plot @8 f @ x D , approx @ x D< , 8 x,- 2.2, 2.2 < , PlotStyle 88 Thickness @ 0.02 D , Blue < , 8 Thickness @ 0.01 D , Red << , PlotRange 8- 0.3, 0.8 < , AxesLabel 8 "x" , "" < , PlotLabel "Powers of x centered at x = 0" D- 2- 1 1 2 x- 0.2 0.2 0.4 0.6 0.8 Powers of x centered at x = Again: Clear @ approx D ; approx @ x _ D = Normal @ Series @ f @ x D , 8 x, 0, 18 <DD ; Plot @8 f @ x D , approx @ x D< , 8 x,- 2.2, 2.2 < , PlotStyle 88 Thickness @ 0.02 D , Blue < , 8 Thickness @ 0.01 D , Red << , PlotRange 8- 0.3, 0.8 < , AxesLabel 8 "x" , "" < , PlotLabel "Powers of x centered at x = 0" D- 2- 1 1 2 x- 0.2 0.2 0.4 0.6 0.8 Powers of x centered at x = No matter what you do, you seem to run up against a barrier near x = - 2 and x = + 2. Go ahead and play by using more and more of the expansion. What piece of advanced mathematics explains why these barriers are mathematical realities? Answer: The key to determining the exact location of the barriers and to learning why they are not figments of your imagination comes from singularities and complex numbers. The function is f @ x D = 1 4- 2x + x 2 . The function has a complex singularity when its denominator is 0: Clear @ z D ; Denominator @ f @ z DD 4- 2 z + z 2 singularities = Simplify @ Solve @ Denominator @ f @ z DD == 0, z DD 99 z 1- 3 = , 9 z 1 + 3 == The complex singularities are 1 + i 3 and 1- i 3 . Plot the points 9 1, 3 = and 9 1,- 3 = together with the center of the plotting interval 8 0, 0 < : sing1 = : 1, 3 > ; sing2 = : 1,- 3 > ; center = 8 0, 0 < ; points = 8 Graphics @8 Red, PointSize @ 0.04 D , Point @ sing1 D<D , Graphics @8 Red, PointSize @ 0.04 D , Point @ sing2 D<D , Graphics @8 Blue, PointSize @ 0.04 D , Point @ center D<D< ; labels = 8 Graphics @ Text @ "sing" , sing1, 8- 1.5, 0 <DD , Graphics @ Text @ "sing" , sing2, 8- 1.5, 0 <DD , Graphics @ Text @ "center" , center, 8- 1,- 1 <DD< ; Show @ points, labels, Axes True, PlotRange All D sing sing center 0.2 0.4 0.6 0.8 1.0- 1.5- 1.0- 0.5 0.5 1.0 1.5 Run line segments from the center of the plotting interval to each singularity:...
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3.05.Convergence - Approximation Authors: Bill Davis,...

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