Math 1B Section 112 Quiz 6 Thursday 4 October 2007 GSI Theo Johnson Freyd http math berkeley edu theojf Name 1 True or False 1 pt each For each of the following statements decide if it is true or false You do not need to show work I will grade only your answers a If a sequence an n 1 is strictly increasing and there s a number M bounding the sequence from above i e an M for every n then limn an exists True This is the an equivalent statement of the monotonic sequences theorem b Let f x be a function and define the sequence an f n If limn an L then limx f x L False It s true that if limx f x exists then this limit equals limn an which necessarily converges But just because a sequence converges does not mean that the function converges e g f x sin x c A geometric series converges if and only if the ratio between successive terms is positive False The ratio must be strictly more than 1 and strictly less than 1 1 2 3 pts Use the divergence test to show that the following series diverges You will need to actually compute a limit or explain why the limit is not defined X n 1 n3 2n3 1 3 1 4 We use the divergence test the limit limn 2nn3 1 limn 21 2n 3 1 1 2 1 1 2 limn 2n3 1 2 0 6 0 Thus since the limit is not 0 the series necessarily diverges 3 4 pts Sum the following telescoping series X n 1 X n 1 3 3 3 3 3n 2 3n 1 4 28 70 3 3n 2 3n 1 1 1 3n 2 3n 1 3 3n 2 3n 1 1 1 1 1 1 1 1 4 4 7 7 10 1 2
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