M147A2 - MATH 147 Assignment #2 Due: Friday, October 9 1)...

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MATH 147 Assignment #2 Due: Friday, October 9 1) For each of the following sequences determine if it converges or diverges. If it converges find the limit: a) { n n +1 } b) { sin( n ) n } c) { n + 1 - n } d) e n n ! e) { n cos( ) 2 n +1 } 2) We will later be able to show that 1 n + 1 < ln( n + 1) - ln( n ) < 1 n a) Let a n = 1 + 1 2 + 1 3 + ··· + 1 n - ln( n ). Prove that { a n } converges. Note: γ = lim n →∞ a n is called Euler’s constant. It is still not known whether or not γ is rational. b) Show that if b n = 1 + 1 2 + 1 3 + ··· + 1 n , then ln( n + 1) < b n ln( n ) + 1. c) Estimate how many terms it would take before b n > 10 6 . How long do you think it would it take for a modern computer to perform this many additions? 3) a) Suppose that a n 0 and that lim n →∞ a n = L . Show that lim n →∞ a n = L . (Hint: Do the cases L = 0 and L > 0 separately. When L > 0 show that a n - L = a n - L a n + L . b)
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M147A2 - MATH 147 Assignment #2 Due: Friday, October 9 1)...

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