Calculus Notes 12.9 - Calculus-Stewart Dr. Berg Summer 09...

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Calculus- Stewart Dr. Berg Summer ‘09 Page 1 12.9 12.9 Representing a Function as a Power Series Power Series Based on the Geometric Series Power series give us an alternate representation of many functions. From the geometric series we get a power series representation f ( x ) = 1 1 x = 1 + x + x 2 +…= x k k = 0 for x < 1 . This can be used to generate power series of similar functions. Examples A By replacing x with x 2 we get f ( x ) = 1 1 x 2 = 1 + ( x 2 ) + ( x 2 ) 2 x 2 k k = 0 which converges for x 2 < 1 or x < 1 . Example B If we replace x with x 2 we get f ( x ) = 1 1 + x 2 = 1 1 − − x 2 ( ) = 1 + ( x 2 ) + ( x 2 ) 2 +… = ( 1) k x 2 k k = 0 for x < 1 . Example C If we replace x with x 2 and multiply the quotient by x 2 2 , we get f ( x ) = x 2 2 + x = x 2 2 1 1 ( ) = x 2 2 1 + x 2 + x 2 2 +… = x 2 2 ( k x 2 k k = 0 = ( k 2 k + 1 k = 0 x k + 2 for x < 2 .
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Calculus Notes 12.9 - Calculus-Stewart Dr. Berg Summer 09...

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