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Quiz7 Solutions

# Quiz7 Solutions - MAC 2312 Quiz 7 SOLUTIONS 1 Determine...

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MAC 2312 Quiz 7 March 13, 2012 SOLUTIONS 1. Determine whether the series n =1 ( 1) n (ln n ) /n converges absolutely, converges conditionally or diverges. Solution: Set b n = (ln n ) /n and a n = ( 1) n b n . Observe that b n 0 for all n = 1 , 2 , 3 , · · · . I claim that b n b n +1 for all n . To see this, consider the related function f ( x ) = ln x x , which satisfies f ( x ) = x · 1 x ln x x 2 = 1 x 2 (1 ln x ) . In particular, f ( x ) 0 for all x 3 so f is decreasing on [3 , ). Therefore, b n b n +1 for all n = 3 , 4 , 5 , · · · . Finally, a routine exercise in L’Hospitals rule (the details of which are omitted here) shows that lim x →∞ f ( x ) = 0. Thus, lim n →∞ b n = 0. By the alternating series test, a n converges. It remains to determine whether a n converges absolutely. For this, observe that for n 3, | a n | = ln n n 1 n . Since n =3 1 /n diverges (by p -series), n =3 | a n | also diverges by comparison. There- fore, a n does not converge absolutely.

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Quiz7 Solutions - MAC 2312 Quiz 7 SOLUTIONS 1 Determine...

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