Quiz 8 Solution - Solutions to Quiz 8 Determine whether the following series are absolutely convergent conditionally convergent or divergent Specify the

Quiz 8 Solution - Solutions to Quiz 8 Determine whether the...

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Solutions to Quiz 8 Determine whether the following series are absolutely convergent, conditionally convergent,or divergent. Specify the test used to reach your conclusion.1.Xn=1nn2-sin2(n) 2.(2n)! 3.(-1)nnln(n)n 4.(-1)n f(x) =xx+1is continuousand non-negative forx1. Moreover, limx→∞f(x) = 0, andf(x) =(x+ 1)12x-x(x+ 1)2=1-x2x(x+ 1)2<0 forx >1.Consequently, the series is convergent by the alternating series test.The series does not converge absolutely: Compare the seriesn=1nn+1to the divergentp-seriesn=11n. We havenn+ 11n=nn+ 1-→n→∞16= 0.Thus the limit comparison test implies the divergence ofn=1nn+1.

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