設(shè)a(n)=1/（n+1）+…+1/2n,(少了1/n,多了1/2n)
lim (1+1/n)^n=e,且(1+1/n)^n
1/(n+1)
b(n+1)-b(n)=1/(n+1)-ln(1+1/n)<0
又b(n)=1+1/2+1/3+...+1/n-lnn
>ln2/1+ln3/2+ln4/3+...+ln(1+1/n)-lnn
=ln(n+1)-lnn>0
故lim b(n)=c,c為常數(shù)
由上題a(n)=b(2n)-b(n)+ln(2n)-lnn
lim a(n)=lim b(2n)-lim b(n)+ln2 ---當(dāng)n趨于無(wú)窮大時(shí),lim b(2n)＝lim b(n)＝c
=c-c+ln2
=ln2
--------2n-1
故 lim∑1/n=lim [a(n)+1/n-1/2n]=lim a(n)+lim 1/n-lim 1/2n=ln2+0-0=ln2
-------i=n