11.2.41-eg-1 - x for which |-3 x 1 |< 1 But | 3 x 3 |<...

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Example Consider the geometric series: X n =0 ( - 3) n ( x + 1) n . Determine the interval of convergence and find the sum of the series on this interval. The geometric series X n =0 r n converges to 1 1 - r provided | r | < 1. Therefore X n =0 ( - 3) n ( x + 1) n is a geometric series with r = ( - 3)( x +1) and the series converges for all values of
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Unformatted text preview: x for which | (-3)( x + 1) | < 1. But | 3 x + 3 | < 1 ⇔ -1 < 3 x + 3 < 1 ⇔ -4 < 3 x <-1 ⇔ -4 3 < x <-1 3 , therefore the interval of convegence is ±-4 3 ,-1 3 ² and the series converges to 1 1-(-3 x-3) = 1 4 + 3 x on this interval....
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