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Assignment 6, Math 220 Due: Monday, March 25th, 2002 1 For each of the following series, indicate whether they converge or diverge and give reasons for your answers. P 1 a: ∞ k 2 +k Pk=1 ∞ b: P k=1 k −k 1 c: ∞ k log k Pk=2 d: P ∞ k=1 sin(π/k) ∞ 1 e: k=3 k2 −3k+2 P∞ k! f: k=1 kk 2 For each of the following series, indicate whether they converge or diverge and give reasons for your answers. Which converge absolutely? P (−1)k √ a: ∞ k=1 k P∞ k b: k=1(−1)k k+1 P∞ c: k=1 (−1)k k2k+1 3 −4 P∞ k 2k k 2 d: k=1(−1) k! P P 2 3 If an with an > 0 is convergent, then is an always convergent? Either prove it or give a counterexample. P√ P an always conver4 If an with an > 0 is convergent, then is gent? Either prove it or give a counterexample. P 5 Can youPgive an example of a convergent series xn and a diverP gent series yn such that (xn + yn ) is convergent. P 6 Find an explicit expression for the nth partial sum of ∞ n=2 log(1 − 2 1/n ) to show that this series converges to − log 2. Is this convergence absolute? 7 Show that the series 1 1 1 1 1 1 1 1 1+ − + + − + + − +··· 2 3 4 5 6 7 8 9 is divergent. 1