【简答题】已知数列{a n }中,a 1 =1,a n <a n+1 ,设b n = a n+1 - a n a n+1 ? a n+1 ,S n =b 1 +b 2 +…+b n ,求证: (Ⅰ) b n <2( 1 a n - 1 a n+1 ) ; (Ⅱ)若数列{a n }是公比为q且q≥3的等比数列,则S n <1.
【简答题】已知数列{a n }满足a 1 =1, a n-1 a n = a n-1 +1 1- a n (n∈N * ,n>1). (1)求证:数列 { 1 a n } 是等差数列; (2)求数列{a n a n+1 }的前n项和S n ; (3)设f n (x)=S n x 2n+1 ,b n =f' n (2),求数列{b n }的前n项和T n .