x/(x3+2)∼1/x2asx→ ∞. By thep-test we know that∞11x2dxconverges, so by thelimit comparison test we know that∞1xx3+ 2dxalso converges and by the comparisontest we conclude that the original integral also converges.3.(25 points)For each of the series below determine whether it converges or diverges. Justifyyour answers.(a)∞n=1n2+ 5n(n+ 1)(n+ 2)(n+ 3)This series diverges.For rational functions the highest power dominates as we go toinfinity. So thenth term is asymptotic ton2/n3= 1/nasngoes to infinity. Since∞11ndiverges by the p-test withp= 1, we conclude that the original series diverges by thelimit comparison test.(b)∞n=21nlnnIn this case we use the integral test. Observe thatdxxlnx= ln(lnx)+Cand limt→∞ln(lnt) =∞since limt→∞lnt=∞. Therefore the improper integral∞2dxxlnxdiverges. By theintegral test the series diverges as well.
(c)∞n=1n23nIn this case the ratio or root test works well. For example, with the ratio test we havelimn→∞(n+ 1)23n+1·3nn2= limn→∞n+ 1n213=13Since this ratio is less than 1, the ratio test says that the series converges.