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138assgn6 - MATH 138 Assignment 6 Sequences(part II...

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MATH 138 Assignment 6 Spring 2009 Sequences (part II). Series (part I). Submit all problems marked (*) by 12:30 p.m. on Friday, June 26. 1. ( * ) Let { a n } be a convergent sequence. (a) Show that lim n →∞ a n +1 = lim n →∞ a n . (b) Suppose that a 1 = 1 and a n +1 = 1 1+ a n for n 1 (we are still assuming that { a n } is convergent). Find its limit. 2. Determine whether the sequence is increasing, decreasing or not monotonic. Is the sequence bounded? (a) a n = n ( - 1) n (b) a n = ne - n 3. ( * ) A sequence { a n } is defined by a 1 = 2, a n +1 = 2 + a n . (a) Prove that { a n } is increasing and bounded by 3. Apply the Monotonic Sequence Theorem to show that lim n →∞ a n exists. (b) Find lim n →∞ a n . 4. Same as in problem 3, for the sequence defined by a 1 = 1, a n +1 = 3 - 1 a n . 5. ( * ) Explain what does it mean to say that n =1 a n = 49. 6. Let a n = 2 n 3 n +1 . (a) Is { a n } a convergent sequence? (b) Is the series a n convergent? 7. Determine whether the following geometric series are convergent or divergent. For the convergent ones, find the sum: (a) ( * ) 1 8 - 1 4 + 1 2 - 1 + . . .
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