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# PP 12.9 - Representations of Functions as Power Series...

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Representations of Functions as Power Series

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Comments 1 if converges it that and series geometric a as viewed be can series that the know We 0 < = x x n n = - - = 1 1 1 that know also We n n r a ar
1 , 1 1 0 1 1 < - = = = = - x x x x n m m n function. for the tion representa series a have then we , 1 1 ) ( if So, x x f - = Of course, this representation is good only for | x | < 1.

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Example e. convergenc of radius its Determine . 2 3 ) ( for tion representa series power a Find x x f - = 1 2 , 2 2 3 1 1 2 3 2 3 0 2 < = - = - = x x x n n x 2 , 2 3 2 3 0 1 < = - = + x x x n n n So, R = 2.
Example . 2 3 ) ( for tion representa series power a Find 2 x x x f - = 2 , 2 3 2 3 0 1 < = - = + x x x n n n 2 , 2 3 2 3 0 1 2 2 < = - = + + x x x x n n n

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Differentiation and Integration of Power Series . 0 radius with e convergenc of interval has ) ( Assume 0 - = R a x c n n n . interval on the ) ( Assume 0 = - = n n n a x c f(x) . ) , ( on able differenti is Then R a R a f(x) + -
) , ( on ) ( ... ) ( 2 ) ( ) ( Moreover, 1 1 2 1 0 R a R a a x nc a x c c a x c dx

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