lec 26 taylor and maclaurin series-2x3

lec 26 taylor and maclaurin series-2x3 - 1 1-x = ∞ X n =0...

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Taylor and Maclaurin Series Per-Olof Persson [email protected] Department of Mathematics University of California, Berkeley Math 1B Calculus Taylor and Maclaurin Series Taylor series of f at a : f ( x ) = X n =0 f ( n ) ( a ) n ! ( x - a ) n = f ( a ) + f 0 ( a ) 1! ( x - a ) + f 00 ( a ) 2! ( x - a ) 2 + f 000 ( a ) 3! ( x - a ) 3 + ··· Maclaurin series if a = 0 : f ( x ) = X n =0 f ( n ) (0) n ! x n = f ( a ) + f 0 (0) 1! x + f 00 (0) 2! x 2 + f 000 (0) 3! x 3 + ··· Remainders Theorem If f ( x ) = T n ( x ) + R n ( x ) , where T n is n th partial sum of the Taylor polynomial of f at a , and lim n →∞ R n ( x ) = 0 for | x - a | < R , then f = sum of its Taylor series on | x - a | < R Theorem (Taylor’s Inequality) If | f ( n +1) ( x ) | ≤ M for | x - a | ≤ d , then the remainder R n ( x ) of the Taylor series satisfies | R n ( x ) | ≤ M ( n + 1)! | x - a | n +1 for | x - a | ≤ d Common Maclaurin Series
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Unformatted text preview: 1 1-x = ∞ X n =0 x n = 1 + x + x 2 + x 3 + ··· R = 1 e x = ∞ X n =0 x n n ! = 1 + x 1! + x 2 2! + x 3 3! + ··· R = ∞ sin x = ∞ X n =0 (-1) n x 2 n +1 (2 n + 1)! = x-x 3 3! + x 5 5!- ··· R = ∞ cos x = ∞ X n =0 (-1) n x 2 n (2 n )! = 1-x 2 2! + x 4 4!- ··· R = ∞ tan-1 x = ∞ X n =0 (-1) n x 2 n +1 2 n + 1 = x-x 3 3 + x 5 5- ··· R = 1 (1 + x ) k = ∞ X n =0 ± k n ² x n = 1 + kx + k ( k-1) 2! x 2 + ··· R = 1...
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