利用级数的性质判定下列级数的敛散性：(1)1+(1/22+1/32)+(1/23+1/33)+(1/24+1/34)+…+(1/2n+1/3n)+…；
(2)(1+1/2)+(3+1/23)+(5+1/25)+(7+1/27)+…；
(3)∑∞n=1(8n/9n+1/n)；
(4)∑∞n=1[(n2+1)/(2n2+n+1)]
(1)因为 sn=1+(1/22+1/32)+…+(1/2n-1+1/3n-1) =(1/2+1/22+…+1/2n-1)+(1/3+1/32+…+1/3n-1)+1/6 =1-1/2n+1/2-1/(3n•2)+1/6． =5/3-1/2n-1/(3n•2)≤5/3． 故该级数收敛． (2)由于limn→∞(2n-1+1/2n-1)=+∞≠0 所以 该级数发散． (3)令un=1/n，Vn=8n/9n+1/n,知un≤Vn(n=1，2，…)且，均为正项级数． 又∑∞n=11/n,发散，故∑∞n=1(8n/9n+1/n,)发散． (4)由于limn→∞[(n+1)/(2n+n+1)]=limn→∞[(1+1/n2)/(2+1/n+1/n2)]=1/2≠0 故该级数发散．
