This series of multiplication of arithmetic progression and geometric progression is arithmetic progression, and the methods used are the same multiplication year-on-year and dislocation subtraction.

sn= 1/2^ 1+3/2^2+……+(2n- 1)/2^n

Multiply by the common ratio 1/2.

1/2 * sn = 1/2^2+……+(2n-3)/2^n+(2n- 1)/2^(n+ 1)

minus

1/2*sn= 1/2+2*( 1/2^2+ 1/2^3+……+ 1/2^n)-(2n- 1)/2^(n+ 1)

sn= 1+4*( 1/2- 1/2^n)+(2n- 1)/2^n

Sn=3-(2n+3)/2^n