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Unformatted text preview: 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 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 be a point in I , i.e., x 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 ....
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This note was uploaded on 12/13/2011 for the course MATH 142 taught by Professor Kustin during the Fall '11 term at South Carolina.
 Fall '11
 KUSTIN
 Calculus, Geometric Series, Taylor Series

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