3.03.UseExpansions - p Approximation 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 3.03 Using Expansions BASICS B.1) Expansions in powers of H x - b L and approximations based on them · B.1.a) When you want to study behavior near a point x = b other than x = 0, you can use expansions in powers of H x - b L . What is the expansion of a function f @ x D in powers of H x - b L ? · Answer: Given a function f @ x D , the expansion of f @ x D in powers of H x - b L is a @ 0 D + a @ 1 D H x - b L + a @ 2 D H x - b L 2 + a @ 3 D H x - b L 3 + . . . + a @ k D H x - b L k + a @ k + 1 D H x - b L k + 1 + . . . + . . . where the numbers a @ 0 D , a @ 1 D , a @ 2 D , . . ., a @ k D , a @ k + 1 D , . . . are set so that for every positive integer m, the function f @ x D and the polynomial a @ 0 D + a @ 1 D H x - b L + a @ 2 D H x - b L 2 + a @ 3 D H x - b L 3 + . . . + a @ m D H x - b L m have order of contact m at x = b. · B.1.b.i) Come up with the expansion of f @ x D = 1 1 + x in powers of H x - 2 L . · Answer: Look at: b = 2; Clear @ f, expan3, x D ; f @ x _ D = 1 1 + x ; expan3 @ x _ D = Normal @ Series @ f @ x D , 8 x, b, 3 <DD 1 3 + 2 - x 9 + 1 27 H - 2 + x L 2 - 1 81 H - 2 + x L 3 The polynomial you see above has order of contact 3 with f @ x D at x = 2. You can check that: Clear @ k D ; Table @ 8 D @ f @ x D , 8 x, k <D , D @ expan3 @ x D , 8 x, k <D< , 8 k, 0, 4 <D ê . x -> 2 :: 1 3 , 1 3 > , : - 1 9 , - 1 9 > , : 2 27 , 2 27 > , : - 2 27 , - 2 27 > , : 8 81 , 0 >> Look at this: Normal @ Series @ f @ x D , 8 x, b, 6 <DD 1 3 + 2 - x 9 + 1 27 H - 2 + x L 2 - 1 81 H - 2 + x L 3 + 1 243 H - 2 + x L 4 - 1 729 H - 2 + x L 5 + H - 2 + x L 6 2187 This polynomial has order of contact 6 with f @ x D at x = 2. The denominators are powers of 3 and the signs alternate. The expansion of f @ x D = 1 1 + x in powers of H x - 2 L is 1 3 - x - 2 3 2 + H x - 2 L 2 3 3 - H x - 2 L 3 3 4 + . . . + H - 1 L k H x - 2 L k 3 k + 1 + . . . This isn't worth memorizing. · B.1.b.ii) Come up with the expansion of f @ x D = Cos @ x D in powers of I x - p 2 M . · Answer: Look at: Clear @ f, x D ; f @ x _ D = Cos @ x D ; Normal B Series B f @ x D , : x, p 2 , 3 >FF p 2 - x + 1 6 J - p 2 + x N 3 This polynomial has order of contact 3 with f @ x D at x = p 2 . Now look at: Normal B Series B f @ x D , : x, p 2 , 12 >FF p 2 - x + 1 6 J - p 2 + x N 3 - 1 120 J - p 2 + x N 5 + I - p 2 + x M 7 5040 - I - p 2 + x M 9 362 880 + I - p 2 + x M 11 39 916 800 This polynomial has order of contact 12 with f @ x D at x = p 2 . The expansion of f @ x D = Cos @ x D in powers of I x - p 2 M is: - I x - p 2 M + I x - p 2 M 3 3 ! - I x - p 2 M 5 5 ! + . . . + H - 1 L k + 1 I x - p 2 M 2k + 1 H 2 k + 1 L ! + . . . which just happens to be the expansion of - Sin A x - p 2 E in powers of I x - p 2 M : Series B - Sin B x - p 2 F , : x, p 2 , 7 >F - J x - p 2 N + 1 6 J x - p 2 N 3 - 1 120 J x - p 2 N 5 + I x - p 2 M 7 5040 + O B x - p 2 F 8 Because: Cos @ x D == - Sin B x - p 2 F True Again, none of these expansions are worth memorizing. · B.1.c)
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3.03.UseExpansions - p Approximation Authors: Bill Davis,...

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