已知数列an满足a1=1/2,a(n+1)=an²+an,则1/(a1+1)+1/(a2+1)+1/(a3+1)+...1/(a2013+1)的值所在区域为( )

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已知数列an满足a1=1/2,a(n+1)=an²+an,则1/(a1+1)+1/(a2+1)+1/(a3+1)+...1/(a2013+1)的值所在区域为()已知数列an满足a1=1/2,

已知数列an满足a1=1/2,a(n+1)=an²+an,则1/(a1+1)+1/(a2+1)+1/(a3+1)+...1/(a2013+1)的值所在区域为( )
已知数列an满足a1=1/2,a(n+1)=an²+an,
则1/(a1+1)+1/(a2+1)+1/(a3+1)+...1/(a2013+1)的值所在区域为( )

已知数列an满足a1=1/2,a(n+1)=an²+an,则1/(a1+1)+1/(a2+1)+1/(a3+1)+...1/(a2013+1)的值所在区域为( )
1/a(n+1)=1/(an^2+an)=1/an-1/(an+1)
1/(an+1)= 1/an-1/a(n+1)
1/(a1+1)+1/(a2+1)+...+1/(a2013+1)=(1/a1-1/a2)+(1/a2-1/a3)+...+(1/a2013-1/a2014)
=1/a1 - 1/a2014=2-1/a2014
因为a(n+1)=an^2 +an
所以a(n+1) -an=an^2 >0
所以{an}是递增数列,
而a2=3/4 a3=21/16
当n>3时,an>a3=21/16
所以0