求下列式子的和:1/2+3/2^2+5/2^3+...+2n-1/2^n

题目

求下列式子的和:1/2+3/2^2+5/2^3+...+2n-1/2^n
数学数列求和

答案

S=1/2+3/2^2+...+(2n-1)/2^n
S/2=1/2^2+.+(2n-3)/2^n+(2n-1)/2^(n+1)
相减：
S/2=1/2+2*(1/2^2+1/2^3+...+1/2^n)-(2n-1)/2^(n+1)
=1/2+1/2+1/2^2+...+1/2^(n-1)-(2n-1)/2^(n+1)
=1/2+1/2*[1-1/2^(n-1)]/(1-1/2)-(2n-1)/2^(n+1)
=1/2+1-1/2^(n-1)-(2n-1)/2^(n+1)
=3/2-(2n+3)/2^(n+1)
S=3-(2n+3)/2^n