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10sTaylorPolySeries

# 10sTaylorPolySeries - Commonly Used Taylor Series series 1...

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Commonly Used Taylor Series series when is valid/true 1 1 - x = 1 + x + x 2 + x 3 + x 4 + . . . note this is the geometric series. just think of x as r = X n =0 x n x ( - 1 , 1) e x = 1 + x + x 2 2! + x 3 3! + x 4 4! + . . . so: e = 1 + 1 + 1 2! + 1 3! + 1 4! + . . . e (17 x ) = n =0 (17 x ) n n ! = X n =0 17 n x n n ! = X n =0 x n n ! x R cos x = 1 - x 2 2! + x 4 4! - x 6 6! + x 8 8! - . . . note y = cos x is an even function (i.e., cos( - x ) = + cos( x ) ) and the taylor seris of y = cos x has only even powers. = X n =0 ( - 1) n x 2 n (2 n )! x R sin x = x - x 3 3! + x 5 5! - x 7 7! + x 9 9! - . . . note y = sin x is an odd function (i.e., sin( - x ) = - sin( x ) ) and the taylor seris of y = sin x has only odd powers. = X n =1 ( - 1) ( n - 1) x 2 n - 1 (2 n - 1)! or = X n =0 ( - 1) n x 2 n +1 (2 n + 1)! x R ln (1 + x ) = x - x 2 2 + x 3 3 - x 4 4 + x 5 5 - . . . question: is y = ln(1 + x ) even, odd, or neither? = X n =1 ( - 1) ( n - 1) x n n or = X n =1 ( - 1) n +1 x n n x ( - 1 , 1] tan - 1 x = x - x 3 3 + x 5 5 - x 7 7 + x 9 9 - . . . question: is y = arctan( x ) even, odd, or neither? = X n =1 ( - 1) ( n - 1) x 2 n - 1 2 n - 1 or = X n =0 ( - 1) n x 2 n +1 2 n + 1 x [ - 1 , 1] 1

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Math 142 Taylor/Maclaurin Polynomials and Series Prof. Girardi Fix an interval I in the real line (e.g., I might be ( - 17 , 19)) and let x 0 be a point in I , i.e., x 0 I . Next consider a function, whose domain is I , f : I R and whose derivatives f ( n ) : I R exist on the interval I for n = 1 , 2 , 3 , . . . , N . Definition 1. The N th -order Taylor polynomial for y = f ( x ) at x 0 is: p N ( x ) = f ( x 0 ) + f 0 ( x 0 )( x - x 0 ) + f 00 ( x 0 ) 2!
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10sTaylorPolySeries - Commonly Used Taylor Series series 1...

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