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Unformatted text preview: MAT A26 Lecture 46 1 Differentiating and Integrating Power Se ries proposition 1.1. Let X n =0 a n x n be a real power series with radius of con vergence R , defining a function f ( x ) for  x  < R . Then (a) the power series X n =0 na n x n 1 obtained by differentiating the original series term by term also has radius of convergence R and sums to f ( x ). (b) the power series X n =0 a n x n +1 n + 1 obtained by integrating the original series term by term also has radius of convergence R and sums to R x f ( x ) dx . (This is also true for complex power series, although we have never yet defined the derivative and integral for complex functions. This is another cliffhanger!! It will be covered in the course Complex Variables as well. ) example 1.2. Integrating the equation 1 1 x = X n =0 x n (for  x  < 1) yields Z x dx 1 x = X n =0 x n +1 n + 1 = X n =1 x n n Now Z x dx 1 x = ln(1 x )  x = ln(1 x ), so ln(1 x ) = X n =1 x n n for  x  < 1 1 remark...
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This note was uploaded on 03/04/2012 for the course MATH 10250 taught by Professor Himonas during the Fall '08 term at Notre Dame.
 Fall '08
 HIMONAS
 Calculus, Power Series

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