7、≥√.√y.
8、kn+pØNõo,n=p=N+1
9、,
10、k
11、≥k=n+1k=n+1qP∞N+1≥√1.√122dÜÂñn,?ênuÑ.n=1P∞P∞(3)ü?êanbnÑÂñ,3êN,n≥Nkan≤n=1n=1P∞un≤bn,?êunǑÂñ.n=1P∞P∞′y.é?¿ε>0,ÏǑ?êanbnÂñ,¤±3N>N,n=1n=1nP+pnP+pnP+pnP+p′n>N,
12、ak
13、<ε,
14、bk
15、<ε.,¡,ak≤uk≤k=n+1k=n+1k=n+1k=n+1nP+pnP+pP∞bk.¤±
16、uk
17、<ε.¤±?êunÂñ.k=n+1k=n+1n=1
18、P∞P∞P∞2.®?êanÂñ,q?êbnuÑ,¯?ê(an±bn)´ÄÂñ?n=1n=1n=1P∞P∞P∞:½uÑ.Äbn=±an±(an±bn)ǑÂñ.n=1n=1n=13.äe?ê´ÄÂñ.P∞√√Pn√√√(1)(n+1−n).ÏǑÜ©Ú(k+1−k)=n+1−1,3n→∞n=1k=1vk4,¤±?êuÑ.P∞PnPn1.111111(2)(2n−1)(2n+1)ÏǑ(2k−1)(2k+1)=2(2k−1−2k+1)=2(1−2n+1),n=1k=1k=13n→∞k4,¤±?êÂñ.P∞(4
19、)cos2π.cos2πnÏǑnlim→∞n=1,¤±?êuÑ.n=11P∞√√nn(7)0.0001.ÏǑlim0.0001=1,¤±?êuÑ.n=1n→∞P∞4.?êunÜ©ÚSǑ{Sn}.en→∞{S2n}{S2n+1}ÑÂñÂn=1P∞ñÓ~êA.y²?êunÂñ.n=1y.é?¿ε>0,ÏǑnlim→∞S2n=limn→∞S2n+1=A,¤±3N>0,n>N,
20、S2n−A
21、<ε,
22、S2n+1−A
23、<ε.u´n>N,
24、Sn−A
25、<ε.¤P∞±limSn=A,?êunÂñ.n→∞n
26、=1P∞5.?êunÂñ,un≥un+1≥0(n=1,2,...),y²:limnun=0.n=1n→∞P∞y.é?¿ε>0,ÏǑ?êunÂñ,dÜÂñn,3N,n=1nP+pP2nεn>N,un<2.u´n>N,(2n)u2n≤2uk<ε.k=n+1k=n+1P2n2nP+1(2n+1)u2n+1≤uk+uk<ε.¤±limnun=0.n→∞k=n+1k=n+1S10.21.?Øe?êñÑ5.P∞P∞(1)nπ.nπ112sin4nÏǑnlim→∞2sin4n/2n=π,?ê2nÂñ
27、,¤±?êÂñ.n=1n=1P∞P∞√1.√1√1√1√1(2)2n3+1ÏǑnlim→∞2n3+1/n3=2,?ên3Âñ,¤±?êÂn=1n=1ñ.P∞(3)√1.√1nnÏǑnlim→∞nn=1,¤±?êuÑ.n=1P∞P∞(4)4n.4n11n2+4n−3ÏǑnlim→∞n2+4n−3/n=4,?ênuÑ,¤±?êuÑ.n=1n=1P∞nnn+2nn13n+1−(5)n+2.ÏǑlimn+2/2=lim(1+2)2=n=1(n2+3n+1)2n→∞(n2+3n+1)2nn→∞nP∞−31e2
28、,?ên2Âñ,¤±?êÂñ.n=1P∞n.n1(6)(lnn)lnnÏǑnlim→∞ln[(lnn)lnn/n2]=limn→∞(3−lnlnn)lnn=−∞,¤n=2P∞n11±nlim→∞(lnn)lnn/n2=0.?ên2Âñ,¤±?êÂñ.n=1P∞q1.n11(7)ntan3nÏǑnlim→∞ntan3n=3,dÜ{,?êÂñ.n=12.?Øe?êñÑ5.2P∞5(n+1)55n.n(1)n!ÏǑnlim→∞(n+1)!/n!=0,d{,?êÂñ.n=1P∞n!.n!(2
29、)3n2ÏǑnlim→∞3n2=+∞,?êuÑ.n=1P∞n3n+1·(n+1)!n3·n!.3·n!1−n−1(3)nnÏǑnlim→∞(n+1)n+1/nn=limn→∞3(1+n)=3e>1,dn=1{,?êuÑ.P∞P∞(4)1.111n1+1/nÏǑnlim→∞n1+1/n/n=1,?ênuÑ,¤±?êuÑ.n=1n=1P∞2q2(5)n.nn1(3−1)nÏǑnlim→∞(3−1)n=3,dÜ{,?
