# taylor_poly - (12.12 Taylor Polynomials Partial sums of...

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(12.12) Taylor Polynomials Partial sums of Taylor series are called Taylor polynomials. Here, we study how closely they approximate a function. Recall if f ( x ) = a 0 + a 1 ( x - c ) + a 2 ( x - c ) 2 + a 3 ( x - 3) 3 + a 4 ( x - c ) 4 + · · · then a n = f ( n ) ( c ) n ! . We define the n -th Taylor polynomial T n ( x ) to be the n -th partial sum of the Taylor series, so T n ( x ) = n k =0 a k ( x - c ) k (where a k = f ( k ) ( c ) k ! ). We define the n -th remainder to be R n ( x ) = f ( x ) - T n ( x ). We will study how closely T n ( x ) approximates f ( x ) using Taylor’s inequality : If | f ( n +1) ( x ) | ≤ M for all x in the interval ( c - d, c + d ), then for all x in that interval we have | R n ( x ) | ≤ M ( n + 1)! | x - c | n +1 . Example Let f ( x ) = sin x . Expand f ( x ) as a Taylor series around c = 0. Recall sin x = x - 1 3! x 3 + 1 5! x 5 - 1 7! x 7 + · · · . Sketch the graph of T 5 ( x ) = x - 1 6 x 3 + 1 120 x 5 with the graph of sin x . Notice how close the graphs are near x = 0. Now we ask the question, how close does T 5 ( x ) approximate sin x for x in the interval [ - 0 . 5 , 0 . 5]?

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