已知函数f(x)=(x+3)/(x+1) (x不等于-1),设数列｛an｝满足a1＝1,an+1(n+1是下标)=f(an)

题目

已知函数f(x)=(x+3)/(x+1) (x不等于-1),设数列｛an｝满足a1＝1,an+1(n+1是下标)=f(an)
数列｛bn｝满足bn＝绝对值(an-√3),sn=b1+b2+……+bn,求证：Sn

答案

a1=1,a=f(an)=1+2/(an+1),
∴an>=1.
√3-a=√3-（an+3)/(an+1)=(√3-1）（an-√3）/(an+1),
∴b=bn*(√3-1）/（an+1)<=bn*(√3-1）/2,
b1=√3-1,bn>0,
∴Sn<=b1/[1-(√3-1）/2]=2(√3-1）/（3-√3）=2√3/3.