n_13491 - 11.9 Representations of Functions as Power Series...

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§ 11.9 Representations of Functions as Power Series Differention and Integration of Power Series Theorem 2 If the power series c n ( x - a ) n has radius of convergence R > 0, then the function f defined by f ( x ) = c 0 + c 1 ( x - a ) + c 2 ( x - a ) 2 + · · · = X n =0 c n ( x - a ) n is defferential (and therefore continuous) on the inteval ( a - R, a + R ) and (i) f 0 ( x ) = c 1 + 2 c 2 ( x - a ) + 3 c 3 ( x - a ) 2 + · · · = n =1 nc n ( x - a ) n - 1 (ii) Z f ( d ) dx = C + c 0 ( x - a ) + c 1 ( x - a ) 2 2 + c 2 ( x - a ) 3 3 + · · · = C + X n =0 c n ( x - a ) n +1 n + 1 The radii of convergence of the power series in Equation (i) and (ii) are both R . Note: 1. Equation (i) and (ii) in Theorem 2 can be rewritten in the form (iii) d dx h n =0 c n ( x - a ) n i = n =0 d dx [ c n ( x - a ) n ] (iv) R h n =0 c n ( x - a ) n i dx = n =0 R c n ( x - a ) n dx 2. Although Theorem 2 says that the radius of convergence remains the same when a power series is dfferentiated or integrated, this does NOT mean that the inteval of convergence remains the same. It may happen
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