一道关于数列的题,要求写出过程题目链接 请写出过程

1.
f(x)=√x,A(n+1)=f(An),Bn=lgAn,B1=1/2.
1/2=B1=lgA1,A1=√10
A(n+1)=√An=(An)^(1/2)
={[A(n-1)]^(1/2)}^(1/2)=[A(n-1)]^(1/4)
={[A(n-2)]^(1/2)}^(1/4)=[A(n-2)]^(1/8)
……
={[(A3)]^(1/2)}^[1/2^(n-3)]=[A3]^[1/2^(n-2)]
={[(A2)]^(1/2)}^[1/2^(n-2)]=[A2]^[1/2^(n-1)]
={[(A1)]^(1/2)}^[1/2^(n-1)]=[A1]^(1/2^n)
=[10^(1/2)]^(1/2^n)=10^[1/2^(n+1)]
所以An=10^（1/2^n)；
Bn=lgAn=lg[10^（1/2^n)]=1/2^n.
2.
Cn=nBn=n/2^n
Sn=1/2+2/2^2+3/2^3+……+(n-1)/2^(n-1)+n/2^n
2Sn=2/2+2/2^1+3/2^2+……+(n-1)/2^(n-2)+n/2^(n-1)
两式相减：
Sn=1/2+1/2^1+1/2^2+……+1/2^(n-2)+1/2^(n-1)-n/2^n
=1/2+（1/2）[1/2^(n-1)-1]/[（1/2）-1]-n/2^n
=1/2-1/2^(n-1)+1-n/2^n
=3/2-2/2^n-n/2^n
=3/2-(n+2)/2^n.

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