Question

n^2/2^n-1求和怎么裂项

Answer 1

Sn=1+4/2+9/4+16/8+25/16+...+(n^2)/2^（n-1) (1)
Sn/2=1/2+4/4+9/8+16/16+25/32+...+(n^2)/2^n (2)
(1)- (2)得
Sn/2=1+3/2+5/4+7/8+9/16+...+(2n-1)/2^（n-1) -(n^2)/2^n (3)
Sn/4=1/2+3/4+5/8+7/16+...+(2n-1)/2^n-(n^2)/2^(n+1) (4)
(3)-(4)得
Sn/4=1+2/2+2/4+2/8+2/16+...+2/2^（n-1) -(2n-1)/2^n-(n^2)/2^(n+1)
=1+1+1/2+1/4+1/8+...+1/2^（n-2) -(2n-1)/2^n-(n^2)/2^(n+1)
=1+2-1/2^（n-2) -(2n-1)/2^n-(n^2)/2^(n+1)
=3-4/2^n-(2n-1)/2^n-(n^2)/2^(n+1)
=3-(2n+3)/2^n-(n^2)/2^(n+1)
=3-(4n+6)/2^(n+1)-(n^2)/2^(n+1)
=3-(n^2+4n+6)/2^(n+1)
Sn=12-(n^2+4n+6)/2^(n-1)