Question

求和：S=1/2+3/4+5/8+7/16+.+（2n-1)/2^n

Answer 1

S=1/2+3/4+5/8+7/16+.+（2n-1)/2^n--------------------------------①

S/2= 1/4+3/8+5/16+.+（2n-3)/2^n+(2n-1)/2^(n+1)-------------②

①-②得：S/2=1/2+1/2+1/2^2+.+1/2^(n-1)-(2n-1)/2^(n+1)

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

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

所以：S=3-(2n+3)/2^n