3.02.Expansions - Approximation Authors: Bill Davis,...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Approximation Authors: Bill Davis, Horacio Porta and Jerry Uhl 1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 3.02 Expansions in Powers of x BASICS B.1) The expansion of a function f @ x D in powers of x Given a function f @ x D , the expansion of f @ x D in powers of x is a @ D + a @ 1 D x + a @ 2 D x 2 + a @ 3 D x 3 + ... + a @ k D x k + a @ k + 1 D x k + 1 + ... where the numbers a @ D , a @ 1 D , a @ 2 D , ... , a @ k D , a @ k + 1 D , ... are chosen so that for every positive integer m, the function f @ x D and the polynomial a @ D + a @ 1 D x + a @ 2 D x 2 + a @ 3 D x 3 + ... + a @ m D x m have order of contact m at x = 0. That's quite a pill to swallow, but the idea is not all that hard once you've gotten some experience. B.1.a) Take f @ x D = 1 2 + x . To get the expansion of f @ x D in powers of x, you could try to work out the m th degree polynomial that has order of contact m with f @ x D at x = 0 as you did in the previous lesson on splines. But Mathematica has a built-in instruction that will do this work automatically for you. Here is the second degree polynomial that has order of contact 2 with f @ x D at x = 0: Clear @ f, x D ; f @ x _ D = 1 2 + x ; Normal @ Series @ f @ x D , 8 x, 0, 2 <DD 1 2- x 4 + x 2 8 Here is the third degree polynomial that has order of contact 3 with f @ x D at x = 0: Normal @ Series @ f @ x D , 8 x, 0, 3 <DD 1 2- x 4 + x 2 8- x 3 16 Here is the fourth degree polynomial that has order of contact 4 with f @ x D at x = 0: Normal @ Series @ f @ x D , 8 x, 0, 4 <DD 1 2- x 4 + x 2 8- x 3 16 + x 4 32 Here is the tenth degree polynomial that has order of contact 10 with f @ x D at x = 0: Normal @ Series @ f @ x D , 8 x, 0, 10 <DD 1 2- x 4 + x 2 8- x 3 16 + x 4 32- x 5 64 + x 6 128- x 7 256 + x 8 512- x 9 1024 + x 10 2048 Look at the pattern emerging above, and give the expansion of f @ x D = 1 2 + x in powers of x. Answer: Look again: Table @ Normal @ Series @ f @ x D , 8 x, 0, k <DD , 8 k, 3, 6 <D TableForm 1 2- x 4 + x 2 8- x 3 16 1 2- x 4 + x 2 8- x 3 16 + x 4 32 1 2- x 4 + x 2 8- x 3 16 + x 4 32- x 5 64 1 2- x 4 + x 2 8- x 3 16 + x 4 32- x 5 64 + x 6 128 Those denominators are consecutive powers of 2, and the signs alternate. The expansion of f @ x D = 1 H 2 + x L in powers of x is: 1 2- x 2 2 + x 2 2 3- x 3 2 4 + x 4 2 5 + ... + H- 1 L k x k 2 k + 1 + ... This is a @ D + a @ 1 D x + a @ 2 D x 2 + a @ 3 D x 3 + ... + a @ k D x k + a @ k + 1 D x k + 1 + ... with a @ k D = H- 1 L k 2 k + 1 . Try it out: Clear @ a, k D ; a @ k _ D = H- 1 L k 2 k + 1 ; Table A Sum A a @ k D x k , 8 k, 0, m <E , 8 m, 4, 6 <E TableForm 1 2- x 4 + x 2 8- x 3 16 + x 4 32 1 2- x 4 + x 2 8- x 3 16 + x 4 32- x 5 64 1 2- x 4 + x 2 8- x 3 16 + x 4 32- x 5 64 + x 6 128 Compare: Table @ Normal @ Series @ f @ x D , 8 x, 0, m <DD , 8 m, 4, 6 <D TableForm 1 2- x 4 + x 2 8- x 3 16 + x 4 32 1 2- x 4 + x 2 8- x 3 16 + x 4 32- x 5 64 1 2- x 4 + x 2 8- x 3 16 + x 4 32- x 5 64 + x 6 128 It checks....
View Full Document

Page1 / 9

3.02.Expansions - Approximation Authors: Bill Davis,...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online