3.06.PowerSeries - Approximation 5 Authors Bill Davis...

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Approximation Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 3.06 Power Series BASICS B.1) Functions defined by power series · B.1.a) What is a power series? Why are power series big deals? · Answer: Any expansion of a function in powers of H x - b L is a power series. Sometimes you use power series to come up with the expansion of a function without having your hands on a formula for the function. In these cases, the function is defined by its expansion. For instance, if the power series you see is 1 + x + x 2 + x 3 + x 4 + ... + x k + ... , then you recognize this power series as the expansion of f @ x D = 1 1 - x in powers of x. On the other hand, if the power series you see is 1 + x + x 2 2 2 + x 3 3 2 + x 4 4 2 + ... + x k k 2 + ... , then you recognize this power series as the expansion of a function f @ x D , but you probably don't know a clean formula for f @ x D . In this case, the best you can say is that f @ x D is defined by this power series. · B.1.b) One function that is defined by a power series is the function f @ x D , whose expansion in powers of x is 1 - x + x 2 2 2 - x 3 3 2 + x 4 4 2 + ... + H - 1 L k x k k 2 + ... What information about f @ x D can you glean from this power series? · Answer: You can get an idea of how f @ x D plots out on short intervals centered at 0. Look at this: Clear @ expan, x, m, k D ; expan @ x _ , m _ D : = 1 + Sum B H - 1 L k x k k 2 , 8 k, 1, m <F The expansion of f @ x D in powers of x through the x 8 term is: expan @ x, 8 D 1 - x + x 2 4 - x 3 9 + x 4 16 - x 5 25 + x 6 36 - x 7 49 + x 8 64 The expansion of f @ x D in powers of x through the x 9 term is: expan @ x, 9 D 1 - x + x 2 4 - x 3 9 + x 4 16 - x 5 25 + x 6 36 - x 7 49 + x 8 64 - x 9 81 Now look at these plots of the expansions of f @ x D through the x 8 and x 9 terms: Plot @8 expan @ x, 8 D , expan @ x, 9 D< , 8 x, - 1.5, 1.5 < , PlotStyle Ø 88 Thickness @ 0.02 D , Blue < , 8 Thickness @ 0.01 D , Red << , AxesLabel Ø 8 "x" , "" <D - 1.5 - 1.0 - 0.5 0.5 1.0 1.5 x 1 2 3 4 5 And look at the plots of the expansions of f @ x D through the x 12 and x 13 terms: Plot @8 expan @ x, 12 D , expan @ x, 13 D< , 8 x, - 1.5, 1.5 < , PlotStyle Ø 88 Thickness @ 0.02 D , Blue < , 8 Thickness @ 0.01 D , Red << , AxesLabel Ø 8 "x" , "" <D - 1.5 - 1.0 - 0.5 0.5 1.0 1.5 x 1 2 3 4 5 Fairly strong evidence of barriers near x = - 1 and x = 1. It seems fairly safe to say that a reasonably trustworthy plot of f @ x D is: Plot @ expan @ x, 13 D , 8 x, - 0.8, 0.8 < , PlotStyle Ø 88 Thickness @ 0.01 D , Blue << , AxesLabel Ø 8 "x" , "f @ x D " <D - 0.5 0.5 x 1.0 1.5 2.0 f @ x D Think of it: All this information with no formula for f @ x D . · B.1.c)
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3.06.PowerSeries - Approximation 5 Authors Bill Davis...

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