# lecture-43 - Lecture 43: Multiplication and Division of...

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Lecture 43: Multiplication and Division of Power Series Dan Sloughter Furman University Mathematics 39 May 20, 2004 43.1 Multiplication of power series The following generalization of the power rule is known as Leibniz’s rule . Theorem 43.1. If f and g are n times diﬀerentiable at z , then d n dz n f ( z ) g ( z ) = n X k =0 ± n k ² f ( k ) ( z ) g ( n - k ) ( z ) . Proof. When n = 1, the result is just the product rule: d dz f ( z ) g ( z ) = f ( z ) g 0 ( z ) + f 0 ( z ) g ( z ) . Assuming the result is true for n 1, we have d n +1 dz n +1 f ( z ) g ( z ) = d dz n X k =0 ± n k ² f ( k ) ( z ) g ( n - k ) ( z ) = n X k =0 ± n k ² ( f ( k ) ( z ) g ( n - k +1) ( z ) + f ( k +1) ( z ) g ( n - k ) ( z ) ) = f ( z ) g ( n +1) ( z ) + n X k =1 ±± n k ² + ± n k - 1 ²² f ( k ) ( z ) g n - k +1 ( z ) + f ( n +1) ( z ) g ( z ) 1

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= n +1 X k =0 ± n + 1 k ² f ( k ) ( z ) g n - k +1 ( z ) , which is the result for the ( n + 1)st derivative. Now suppose
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## This note was uploaded on 11/23/2009 for the course MATHEMATIC Mathematic taught by Professor Dansloughter during the Spring '04 term at Furman.

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lecture-43 - Lecture 43: Multiplication and Division of...

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