# Lecture 20 - Series Solutions of ODEs and Taylor...

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Series Solutions of ODEs and Taylor Polynomials

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MATLAB Quiz Thursday, March 13 Same Time As Your Section (APM B432) Alternate Time Sign Ups Are Online DON’T FORGET!!!! (NO MAKEUPS)
Taylor Series For a Power Series a n ( x - x 0 ) n 1 X n =0 = f ( x ) a n It is always true that = f ( n ) ( x 0 ) n ! So given a function, we can write it as a power series. A power series that describes a function is called a “Taylor Series”

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Taylor Series For a Power Series a n ( x - x 0 ) n 1 X n =0 = f ( x ) For Example e x = 1 + x + x 2 2 ! + x 3 3 ! + ... 1 X n =0 x n n ! =
Taylor Series e x 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 e x = 1 + x + x 2 2 ! + x 3 3 ! + ... 1 X n =0 x n n ! =

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Taylor Series e x 1 + x 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 e x = 1 + x + x 2 2 ! + x 3 3 ! + ... 1 X n =0 x n n ! =
Taylor Series e x 1 + x + x 2 2 ! 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 e x = 1 + x + x 2 2 ! + x 3 3 ! + ... 1 X n =0 x n n ! =

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Taylor Series 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 e x 1 + x + x 2 2 !
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• Winter '06
• Mohanty

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