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Unformatted text preview: Taylor Series, Taylor Polynomials John E. Gilbert, Heather Van Ligten, and Benni Goetz In the previous lecture we started with a power series like n a n x n or n a n ( x- c ) n and used it to define a function f ( x ) . Now we want to reverse the process by starting with a function f ( x ) and trying to represent it as a power series f ( x ) = n a n x n or f ( x ) = n a n ( x- c ) n centered at some point x = c . But whats a n in this case? The answer is contained in one of the most fundamental results in single variable calculus: Definition: for a function f the Taylor Series of f centered at x = c is f ( x ) = f ( c ) + f ( c )( x- c ) + f ( c ) 2! ( x- c ) 2 + . . . + f ( n ) ( c ) n ! ( x- c ) n + . . . . When the Taylor series is centered at the origin it becomes f ( x ) = f (0) + f (0) x + f (0) 2! x 2 + . . . + f ( n ) (0) n ! x n + . . . = summationdisplay n =0 f ( n ) n ! x n , and it is then often called the Maclaurin Series of f . Technically speaking, there are functions f where the Taylor series does not converge to f ( x ) , but these rarely occur in practice, so we shall simply assume that when a Taylor series converges, it converges to f ( x ) and speak of the Taylor series representation of f ( x ) . Before trying to understand why the coefficients are the way they are, lets look at some already familiar examples: Example 1: f ( x ) = 1 1- x Solution: since f ( x ) = 1 1- x = f ( x ) = 1 (1- x ) 2 , while f ( x ) = 2 (1- x ) 3 , f (3) (...
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