Question

已知数列{an}满足an=n/2的n次方,求Sn

Answer 1

答：

an=n/2^n

Sn=a1+a2+a3+…+an

=1/2+2/2^2+3/2^3+…+n/2^n

2Sn=1+2/2+3/2^2+…+n/2^(n-1)

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

=1+1/2+1/2^2+1/2^3+…+1/2^(n-1)-n/2^n

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

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