An为等差数列,Bn是各项都为正数的等比数列,An=1+(n-1)d=2n-1,Bn=2的n次方,求数列An/Bn的前n
An为等差数列,Bn是各项都为正数的等比数列,An=1+(n-1)d=2n-1,Bn=2的n次方,求数列An/Bn的前n项和Sn.
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An/Bn=(2n-1)/2^n
Sn=S1+S2+S3+……+Sn=(2*1-1)/2^1+(2*2-1)/2^2+(2*3-1)/2^3+……(2*n-1)/2^n=(1/2^0-1/2^1)+(2/2^1-1/2^2)+(3/2^2-1/2^3)+……+(n/2^(n-1)-1/2^n)=1/2^0+2/2^1+3/2^2+……+n/2^(n-1)-(1/2^1+1/2^2+……+1/2^n)
令Tn=1/2^0+2/2^1+3/2^2+……+n/2^(n-1)
Pn=1/2^1+1/2^2+……+1/2^n
则Sn=Tn-Pn
Pn=[1/2(1-1/2^n)]/(1-1/2)=1-1/2^n
1/2Tn=1/2^1+2/2^2+3/2^3+……+n/2^n
Tn-1/2Tn=1/2^0+1/2^1+……+1/2^(n-1)-n/2^n
1/2Tn=[1/2^0(1-1/2^n)]/(1-1/2)-n/2^n
Tn=4-4/2^n-2n/2^n
则Sn=Tn-Pn=4-4/2^n-2n/2^n-(1-1/2^n)=3-(3+2n)/2^n
Sn=S1+S2+S3+……+Sn=(2*1-1)/2^1+(2*2-1)/2^2+(2*3-1)/2^3+……(2*n-1)/2^n=(1/2^0-1/2^1)+(2/2^1-1/2^2)+(3/2^2-1/2^3)+……+(n/2^(n-1)-1/2^n)=1/2^0+2/2^1+3/2^2+……+n/2^(n-1)-(1/2^1+1/2^2+……+1/2^n)
令Tn=1/2^0+2/2^1+3/2^2+……+n/2^(n-1)
Pn=1/2^1+1/2^2+……+1/2^n
则Sn=Tn-Pn
Pn=[1/2(1-1/2^n)]/(1-1/2)=1-1/2^n
1/2Tn=1/2^1+2/2^2+3/2^3+……+n/2^n
Tn-1/2Tn=1/2^0+1/2^1+……+1/2^(n-1)-n/2^n
1/2Tn=[1/2^0(1-1/2^n)]/(1-1/2)-n/2^n
Tn=4-4/2^n-2n/2^n
则Sn=Tn-Pn=4-4/2^n-2n/2^n-(1-1/2^n)=3-(3+2n)/2^n
1年前
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