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Calculus 2.qxd12/6/071:46 PMPage 1BarCharts, Inc.Series continued3/ a n =S means the series convergesand its sum is S. In general statements, / a n may stand for3/ a n =S.An equation such asn=0n=0 Geometric series. A (numerical) geometric series has the3form/ ar n, where r is a real number and a 0. A key identityn=0NnN+1] r!1 g . It impliesis / r =1+r +r2 +...+r N=1-r1-rn =01 ^ if r <1 h e also / ar n =a b 1 -1 lo,/ r n = 1r1-r33n=0n=1andthat the series diverges if |r|>1. The series diverges ifr= 1. The convergence and possible sum of any geometricseries can be determined using the preceding formula.The p-series and geometric series are often used forcomparisons. Try a limit comparison when a serieslooks like a p-series, but is not directly comparable to it.3sin ^ 1/n 2 hE.g., / sin ^ 1/n 2 h converges since lim=1.n"31/n 2n=K Ratio & root tests. Assume an 0.a n+1If lim a <1 or lim a n 1/n <1, then / a n convergesn"3n"3na n+11/n(absolutely). If lim a >1 or lim a n >1, thenn"3n"3n/ a n diverges. These tests are derived by comparison withgeometric series. The following are useful in

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