求：Sn=2/2+3/2^2+4/2^3+…+n/2^n-1+n+1/2^n

来源：百度知道编辑：UC知道时间：2024/06/23 01:46:27

谢谢

Sn=2/2+3/2^2+4/2^3+…+n/2^(n-1)+(n+1)/2^n...........(1)
2Sn=2+3/2+4/2^2+5/2^3+...+n/2^n-2+(n+1)/2^(n-1)......(2)
(2)-(1)
Sn=2+1/2+1/2^2+1/2^3+...+1/2^(n-1)-(n+1)/2^n.
=2+1-1/2^n-1-n+1/2^n.
=3-2/2^n-(n+1)/2^n.
=3-(n+3)/2^n

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

所以Sn=3-2^(-n)-(n+1)*2(-n)
Sn=3-(n+2)2^(-n)

2Sn-Sn

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