SeriesExpansions

# SeriesExpansions - Inverse Tan Function tan-1 x = x-x 3 3 x...

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SERIES EXPANSIONS (1) Taylor series: (i) centered at origin f ( x ) = f (0) + f 0 (0) x + f 00 (0) 2! x 2 + . . . = X n = 0 f ( n ) (0) n ! x n . (ii) centered at x = a f ( x ) = f ( a ) + f 0 ( a )( x - a ) + f 00 ( a ) 2! ( x - a ) 2 + . . . = X n = 0 f ( n ) ( a ) n ! ( x - a ) n . (2) degree N Taylor Polynomial: (i) centered at origin T N ( x ) = f (0) + f 0 (0) x + . . . + f ( N ) (0) N ! x n = N X n = 0 f ( n ) (0) n ! x n . (ii) centered at x = a T N ( x ) = N X n = 0 f ( n ) ( a ) n ! ( x - a ) n . EXAMPLES (3) Geometric Series 1 1 - x = 1 + x + x 2 + x 3 + . . . = X n = 0 x n . (4) Exponential Function: e x = 1 + x + x 2 2! + . . . = X n = 0 x n n ! . (5) Log Function: ln(1 + x ) = x - x 2 2 + x 3 3 - . . . = X n = 1 ( - 1) n - 1 n x n . (6)
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Unformatted text preview: Inverse Tan Function: tan-1 x = x-x 3 3 + x 5 5-. . . = ∞ X n = 1 (-1) n-1 2 n-1 x 2 n-1 . (7) Sin Function: sin x = x-x 3 3! + x 5 5! + . . . = ∞ X n = 1 (-1) n-1 (2 n-1)! x 2 n-1 . (8) Cos Function: cos x = 1-x 2 2! + x 4 4!-. . . = ∞ X n = 1 (-1) n (2 n )! x 2 n . 1...
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