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Lecture46-rev

# Lecture46-rev - MAT A26 Lecture 46 1 Differentiating and...

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MAT A26 Lecture 46 1 Differentiating and Integrating Power Se- ries proposition 1.1. Let 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 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 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 x 0 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 = n =0 x n (for | x | < 1) yields x 0 dx 1 - x = n =0 x n +1 n + 1 = n =1 x n n Now x 0 dx 1 - x = - ln(1 - x ) | x 0 = - ln(1 - x ), so ln(1 - x ) = - n =1 x n n for | x | < 1 1

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remark 1.3. It is tempting to “prove” 1 - 1 2 + 1 3 - 1 4 + · · · = ln 2 by writing ln 2 = lim x → - 1 ln(1 - x ) = - lim x → - 1 n =1 x
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Lecture46-rev - MAT A26 Lecture 46 1 Differentiating and...

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