Solutions to Quiz 8Determine 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)n4.(-1)nf(x) =√xx+1is continuousand non-negative forx≥1. Moreover, limx→∞f(x) = 0, andf(x) =(x+ 1)12√x-√x(x+ 1)2=1-x2√x(x+ 1)2<0 forx >1.Consequently, the series is convergent by the alternating series test.The series does not converge absolutely: Compare the series∑∞n=1√nn+1to the divergentp-series∑∞n=11√n. We have√nn+ 11√n=nn+ 1-→n→∞16= 0.Thus the limit comparison test implies the divergence of∑∞n=1√nn+1.