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3.02.Expansions

# 3.02.Expansions - 1 2 1 2 Approximation x 4 x 4 x2 8 x2 8...

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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 @ 0 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 @ 0 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 @ 0 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 êê 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 êH 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 @ 0 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 êê 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 êê 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. B.2) The expansions every literate calculus person knows: Æ The expansion of 1 1 - x in powers of x is 1 + x + x 2 + x 3 + x 4 + ... + x k + ... Æ The expansion of e x in powers of x is 1 + x + x 2 ë 2 + x 3 ë 3 ! + ... + x k ë k ! + ... Æ The expansion of Sin @ x D in powers of x is x - x 3 ë 3 ! + x 5 ë 5 ! + ... + H - 1 L k x 2 k + 1 ë H 2 k + 1 L ! + .. .

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