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PowerSeries

# PowerSeries - Taylor Polynomials and Power Series Taylor...

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Taylor Polynomials and Power Series Taylor Polynomials and Taylor Series. Suppose that f has N derivatives at a. Then let P N ( x ) = f ( a ) + ! f ( a )( x " a ) + !! f ( a ) 2 ( x " a ) 2 + ! + f ( N ) ( a ) N ! ( x " a ) N This is the N th Taylor polynomial for f centered at a . If a = 0 , then it is known as the N th Maclaurin polynomial for f. P N ( x ) = f (0) + ! f (0) x + !! f (0) 2 x 2 + ! + f ( N ) (0) N ! x N It is easy to see that f ( n ) ( a ) = P N ( n ) ( a ) for 0 ! n ! N So, we have created a polynomial whose derivatives agree with that of f at a up to N . A Taylor series is obtained by taking the limit. So, the Taylor series for f centered at a is given by the following. T ( x ) = f ( n ) ( a ) n ! ( x ! a ) n n = 0 " # By this means we have defined a new function such that T ( n ) ( a ) = f ( n ) ( a ) for n = 0,1,2, . There are two questions that we want to answer concerning these Taylor series. For which values of x do they converge? For which values of x do they represent the original function that was used to define them? Give a function f that is infinitely differentiable at a point a , we can define the Taylor series quite easily. Here are a few examples. f ( x ) = 1 1 ! x a = 0 T ( x ) = x n n = 0 ! " f ( x ) = 1 1 + x a = 0 T ( x ) = ( ! 1) n x n n = 0 " # f ( x ) = ln(1 + x ) a = 0 T ( x ) = ( ! 1) n x n + 1 n + 1 n = 0 " #

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f ( x ) = arctan( x ) a = 0 T ( x ) = ( ! 1) n x 2 n + 1 2 n + 1 n = 0 " # f ( x ) = e x a = 0 T ( x ) = x n n ! n
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PowerSeries - Taylor Polynomials and Power Series Taylor...

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