3.06.PowerSeries

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

This preview shows pages 1–2. Sign up to view the full content.

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)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 15

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online