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11.2.7-eg-1

# 11.2.7-eg-1 - Example Consider the following two innite...

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Example Consider the following two infinite series: (a) X n =0 2 9 n (b) X n =1 3 n - 1 ( - 7) n . For each series, find its sum and tell whether it converges or diverges. Solution: Each of the series in (a) and (b) is a geometric series. Recall that a geometric series may be written as X n =0 ar n or X n =1 ar n - 1 (the two summations are equivalent), and that if | r | < 1, the series converges with sum given by X n =0 ar n = X n =1 ar n - 1 = a 1 - r . Otherwise, if | r | ≥ 1, the geometric series diverges. (a) X n =0 2 9 n Because the summation for this series begins with the n = 0 term, it is natural to try to express it in the form X n =0 ar n . Since X n =0 2 9 n = X n =0 2 · 1 9 n , it must be that a = 2 , r = 1 9 , so that | r | < 1. Thus the series converges and its sum is given by X n =0 2 · 1 9 n = 2 1 - 1 9 = 2 8 9 = 18 8 = 9 4 .

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11.2.7-eg-1 - Example Consider the following two innite...

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