Question

数列放缩题.放不到正好是5/3啊。

Answer 1

Sn = 2n-an

n=1,a1= 1

an = Sn -S(n-1)

= 2-an +a(n-1)

an - 2 = (1/2)[a(n-1) -2 ]

{an -2} 是等比数列,q=1/2

an -2 = (1/2)^(n-1) .(a1-2)

an = 2-(1/2)^(n-1)

bn= 2^(n-1).an

=2^n -1

consider

for n>=3

1/(2^n-1) < (2/3) (1/2)^(n-1)

1/b1+1/b2+...+1/bn

=1/b1+1/b2+(1/b3+...+1/bn)

=1+1/3+[1/(2^3-1) +...+1/(2^n-1) ]

< 4/3 + (2/3)[ 1/2^2+...+1/2^(n-1) ]

=4/3 + (1/3)( 1- 1/2^(n-3) )