计算1+3/2+5/2^2+7/2^3+……+2*1993+1/2^1993=()A、6-3991/2^1993B、3-3991/2^1994C、6-3991/2^1994D、3-3991/2^1993

Question

计算1+3/2+5/2^2+7/2^3+……+2*1993+1/2^1993=( )
A、6-3991/2^1993
B、3-3991/2^1994
C、6-3991/2^1994
D、3-3991/2^1993

Answer 1

设原式为S,有
S = 1+3/2+5/2^2+7/2^3+……+(2×1993+1)/2^1993
1/2S = 1/2+3/2^2+5/2^3+7/2^4+……+(2×1993+1)/2^1994
两式相减得
S - 1/2S = 1+2/2+2/2^2+2/2^3+……+2/2^1993-(2×1993+1)/2^1994
= 1+2(1/2+1/2^2+……+1/2^1993)-(2×1993+1)/2^1994
= 1+2×(1-1/2^1993)-(2×1993+1)/2^1994
= 3 - 3991/2^1994
选B