S 12.1 Definition and Properties of Infinite SeriesIl.Convergence ofinfinite seriesE.g.1 Show that a geometric series80Z= a +aq+aq +...+aq"-1 +..(a+ 0)aq'n=1converges if q < 1, but diverges if q| ≥ 1.Solution:aaqa-aqn-1Sn =a+aq+aq’ +...+ag"1-q1-q1-qalims,when|q| <1,limq"= 0converges1-qn->n->00when q>1,lims,=o0limgn=0divergesn-00n>o
§12.1 Definition and Properties of Infinite Series II. Convergence of infinite series convergesif 1,but diverges if 1. ( 0) Show that a geometric series 2 1 1 1 = + + + + + − = − q q aq a aq aq aq a n n n E.g.1 Solution: 2 −1 = + + + + n sn a aq aq aq q a aqn − − = 1 q aq q a n − − − = 1 1 when 1, q lim = 0 → n n q q a sn n − = → 1 lim when 1, q = → n n limq = → n n lim s converges diverges
S 12.1 Definition and Properties of Infinite SeriesIl.Convergenceofinfinite seriesE.g.1 Show that a geometric series80Z= a+ aq+aq +...+aq"-- +...(a+ 0)aq'n=1converges if q <1, but diverges if q| ≥1.Solution:when q=l, Sn = na →0divergeswhen q=-l, the seriesis, a-a+a-a+.lim s, doesn't exist.divergesn→808MSo,aq"-1 convergesif |al <1, but diverges if || ≥1.n=l
§12.1 Definition and Properties of Infinite Series II. Convergence of infinite series E.g.1 Solution: when 1, q = when 1, q = − sn = na → convergesif 1,but diverges if 1. ( 0) Show that a geometric series 2 1 1 1 = + + + + + − = − q q aq a aq aq aq a n n n the seriesis, a − a + a − a + lim doesn't exist. n n s → diverges diverges So, convergesif 1,but diverges if 1. 1 1 = − aq q q n n