已知a1=1，a=an/(1+an)则，a2=1/2；a3=1/3；a4=1/4……猜想，an=1/n显然满足a=an/(1+an)故，an=1/n已知an=1/n则，an/(n+1)=1/[n(n+1)]=(1/n)-[1/(n+1)]所以，(a1/2)+(a2/3)+(a3/4)+……+(an/n+1)=[1-(1/2)]+[(1/2)-(1/3)]+[(1/3)-(1/4)]+……[(1/n)-(1/n+1)]=1-[1/(n+1)]=n/(n+1)＜1
