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

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Unformatted text preview: Example Consider the following two infinite series: ∞ (a) n=0 ∞ 2 3n−1 (b) 9n 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 ∞ ∞ arn arn−1 or n=0 n=1 (the two summations are equivalent), and that if |r| < 1, the series converges with sum given by ∞ ∞ a n . ar = arn−1 = 1−r n=0 n=1 Otherwise, if |r| ≥ 1, the geometric series diverges. ∞ (a) n=0 2 9n Because the summation for this series begins with the n = 0 term, it is natural to try to ∞ arn . Since express it in the form n=0 n=0 2 = 9n ∞ 2· n=0 1 9 n , it must be that 1 r= , 9 so that |r| < 1. Thus the series converges and its sum is given by a = 2, ∞ 2· n=0 1 9 n = 2 1− 1 9 = 2 8 9 = 18 9 =. 8 4 ∞ (b) n=1 3 n −1 (−7)n . The summation for this series begins with the n = 1 term, so it is natural to try to express ∞ arn−1 . This means that all powers inside the summation need to the series in the form n=1 be of the form (n − 1). We have ∞ n=1 3n−1 = (−7)n ∞ n=1 ∞ = n=1 ∞ 3n−1 (−7)1 · (−7)n−1 1 3n−1 · (−7)1 (−7)n−1 − = n=1 1 7 − 3 7 n−1 ∞ arn−1 with so that the series is now in the form of n=1 1 a=− , 7 3 r=− . 7 Again, |r| < 1 so the geometric series converges, and its sum is given by ∞ n=1 1 − 7 3 − 7 n−1 = −1/7 −1/7 1 = =− . 1 − (−3/7) 10/7 10 2 ...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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

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