series - 1 Any analytic function We can expand around x0 f...

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1 Any analytic function We can expand around x 0 f ( x ) = f ( x 0 ) + f 0 ( x 0 ) δx + f 00 ( x 0 ) ( δx 2 ) 2! + . . . where δx = x - x 0 2 Trig and Exponential sin( x ) = x - x 3 3! + x 5 5! ... cos( x ) = 1 - x 2 2! + x 4 4! ... e x = 1 + x + x 2 2! + ... Show that e ix = cos( x ) + i sin( x ) by comparing power series 3 Power and Log 1 1 + x = 1 - x + x 2 + ... log(1 + x ) = x - x 2 2 + x 3 3 + ... (1 + x ) α = 1 + αx + α ( α - 1) 2! x 2 + ... Show that R x dx 0 1 1+ x 0 = log(1 + x ) by integrating term by term 4 Examples from quizzes 1. For β ± 1: s 1 - β 1 + β 1 - β + O ( β 2 ) 2. For p ± T 1 e p/T - 1 T p + O (1) 3. For p ² T , e p/T ²
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