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TaylorSeries - Taylor Series Taylor Polynomials John E...

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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 what’s 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, let’s look at some already familiar examples:
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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) ( x ) = 2 · 3 (1 - x ) 4 , ...
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