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power_series - (12.8 Power Series Example For what x does...

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(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 that’s 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 | > 2. So it diverges if x < - 2 or x > 2.
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