题目出错了,不等号方向反了,而且没有规定n的取值.
证：
假设当n=k(k∈N,且k≥1)时,ak>0,则当n=k+1时,a(k+1)=ak²+ak=ak(ak+1)
ak>0 ak +1>0 a(k+1)>0
k为任意正整数,因此an>0,即数列各项均>0
a(n+1)=an²+an
a(n+1)/an=an+1>1
a(n+1)>an,数列为递增数列.
a2=a1²+a1=1+1=2
n≥2时,an≥2 1/an≤1/2
a(n+1)=an²+an
1/a(n+1)=1/[an²+an]=1/[an(an +1)]=1/an -1/(an +1)
1/(an +1)=1/an -1/a(n+1)
1/(a1+1)+1/(a2+1)+...+1/(an +1)
=1/a1-1/a2+1/a2-1/a3+...+1/an -1/a(n+1)
=1/a1 -1/a(n+1)
=1 -1/a(n+1)
n≥1,n+1≥2 a(n+1)≥2 0