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Unformatted text preview: (12.8) Power Series Example. For what x does the series n =0 1 3 n x n = 1 + 1 3 x + 1 9 x 2 + 1 27 x 3 + converge? This is just a geometric series. Recall n =0 ar n converges to a 1 r provided  r  < 1. Here, a = 1 and r = x/ 3. So the series converges to 1 1 x/ 3 = 3 3 x provided  x/ 3  < 1, or  x  < 3. That is, the series converges exactly when 3 < x < 3. We can think of this series as a function: f ( x ) = 1+ 1 3 x + 1 9 x 2 + 1 27 x 3 + . This function has domain given by the interval ( 3 , 3) because thats where the series converges. Actually, f ( x ) = 3 3 x , but with domain ( 3 , 3). Example. For what x does the series n =1 x n 2 n n = 1 2 x + 1 8 x 2 + 1 24 x 3 + converge? We can use the ratio test. Here, a n = x n 2 n n . When we do the ratio test, we take absolute values because depending on what x we choose, the series might not be positive. So lim n a n +1 a n = lim n x n +1 2 n +1 ( n + 1) x n 2 n n = lim n x n +1 2 n +1 ( n + 1) 2 n n x n = lim n nx 2( n + 1) =  x  2 So we know the series converges if  x  2 < 1. (In fact, it converges absolutely in that case.) So it converges absolutely if  x  < 2 or 2 < x < 2. We also know that it diverges if  x  2 > 1, or  x  >...
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This note was uploaded on 04/02/2008 for the course MATH 31B taught by Professor Valdimarsson during the Fall '08 term at UCLA.
 Fall '08
 VALDIMARSSON
 Geometric Series, Power Series

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