已知数列1*1/2,2*1/4,3*1/8,4*1/16,···,n*1/2^n,···,求sn
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![已知数列1*1/2,2*1/4,3*1/8,4*1/16,···,n*1/2^n,···,求sn](/uploads/image/z/117590-14-0.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%971%2A1%2F2%2C2%2A1%2F4%2C3%2A1%2F8%2C4%2A1%2F16%2C%C2%B7%C2%B7%C2%B7%2Cn%2A1%2F2%5En%2C%C2%B7%C2%B7%C2%B7%2C%E6%B1%82sn)
已知数列1*1/2,2*1/4,3*1/8,4*1/16,···,n*1/2^n,···,求sn
已知数列1*1/2,2*1/4,3*1/8,4*1/16,···,n*1/2^n,···,求sn
已知数列1*1/2,2*1/4,3*1/8,4*1/16,···,n*1/2^n,···,求sn
设第n项为an
a1=1×1/2^1
a2=1×2/2^2
a3=1×3/2^3
…………
an=n/2^n
Sn=a1+a2+...+an
=1/2^1+2/2^2+3/2^3+...+n/2^n
Sn/2=1/2^2+2/2^3+...+(n-1)/2^n+n/2^(n+1)
Sn-Sn/2=Sn/2=1/2^1+1/2^2+1/2^3+...+1/2^n-n/2^(n+1)
=(1/2)[1-(1/2)^n]/(1-1/2)-n/2^(n+1)
=1-1/2^n-n/2^(n+1)
Sn=2-2/2^n-n/2^n
=2-(n+2)/2^n