Math 1B Section 107 Quiz 8 Thursday 18 October 2007 GSI Theo Johnson Freyd http math berkeley edu theojf Name For the first two questions the alternating series diverge For each series decide which parts of the Alternating Series Test it satisfies and which parts it fails to satisfy 1 2 pts X 1 n n 1 n 1 2n The alternating series test demands that a series be P P a alternating either 1 n an or 1 n 1 an where an 0 b decreasing an an 1 c tending towards 0 limn an 0 This series satisfied a and b but fails to have the correct limit 1 n 1 n 2n 2 lim 1 1 1 1 1 1 1 1 1 1 1 4 3 16 5 64 7 256 9 1024 11 4096 This series satisfies parts a and c above but fails to satisfy part b for example 1 1 1 1 5 16 and 7 64 2 2 pts 1 1 For the next two questions use the Ratio Test to determine if the series converges or diverges n X 1 3 3 pts n 4 n 0 We use the ratio test lim 1 n 1 4 1 n 4 n 1 n n lim n 1 n 1 1 4 so the series diverges 4 3 pts X n 0 2n n 1 We use the ratio test n 1 2n 1 1 n n 2n 1 lim lim n 1 n 1 2n 1 lim n 1 n 2 n 2 1 so the series converges absolutely 2 1 2n 1 2n 1 1 2
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