pow_series_for_funct

pow_series_for_funct - (12.9 Power Series for Functions...

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(12.9) Power Series for Functions Recall the formula for geometric series: ± n =0 ar n = a 1 - r provided | r | < 1. (We really mean, ± n =0 ar n converges to a 1 - r provided | r | < 1.) So if we let a = 1 and r = x , we have ± n =0 x n =1+ x + x 2 + x 3 + ··· = 1 1 - x provided | x | < 1, or - 1 <x< 1. So we think of f ( x )= ± n =0 x n as a function of x . We therefore know is that it is really the function f ( x )= 1 1 - x , but only for - 1 <x< 1 (for x outside that interval, the function f ( x )i s unde±ned). We can use this series to ±nd power series for other functions. Example Find a power series for the function f ( x )= 2 x x - 3 . We start with 1 1 - x =1+ x + x 2 + x 3 + ··· . Trick: substitute x 3 for x . We have 1 1 - x 3 =1+ x 3 + ² x 3 ³ 2 + ² x 3 ³ 3 + ··· or 3 3 - x =1+ x 3 + x 2 9 + x 3 27 + ··· . We adjust this a little (by dividing by 3 then multiplying by - 1): 1 x - 3 = - 1 3 - x 9 - x 2 27 - x 3 81 -··· . This is 1 x - 3 = ± n =0 - x n 3 n +1 . Now multiply by 2 x to get the desired result: 2 x x - 3 = - 2 x 3 - 2 x 2 9 - 2 x 3 27 - 2 x 4 81 -··· = ± n =0 - 2 x n +1 3 n +1 = ± n =0 - 2 x n 3 n .
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pow_series_for_funct - (12.9 Power Series for Functions...

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