S 12.2 Convergence tests for series of constantI.PositiveSeries80181ZE.g.221Zn(n+ 1)In!/4n2+25n=1n=1=1111Analysisn+1/n(n+1)hn(n+1)11110n/4n2+252nV4n2+25111n!1.2.3..n2.2.2...22n
§12.2 Convergence tests for series of constant I. Positive Series E.g.2 4 25 1 1 2 n= n + ! 1 1 ( 1) n= n 1 1 n= n n + Analysis n n n 1 ( 1) 1 + 1 1 ( 1) 1 + n n + n n 2n 1 4 25 1 2 + n 10n 1 4 25 1 2 + n n = 1 2 3 1 ! 1 2 2 2 2 1 n 2 1
S 12.2 Convergence tests for series of constantI.Positive Series8E.g.3>The series2n=1is called a p-series.Show that it converges if p > 1and diverges if p ≤1.Analysis-2If p=1,diverges1If p<1,divergeshpnIf p>1
§12.2 Convergence tests for series of constant E.g.3 and diverges if 1. is called a -series.Show that it convergesif 1 1 3 1 2 1 1 1 The series 1 = + + + + + = p p p n n p p p n p I. Positive Series Analysis = = = = 1 1 1 1 If 1, n n p n n p n n p p 1 1 If 1, diverges diverges If p 1