神马作文网2^n/n*(n+1)
来源：学生作业帮 编辑：神马作文网作业帮 分类：数学作业 时间：2024/11/23 23:49:32
2^n/n*(n+1)
1/2*f(1/2)=(1/2)^2+3*(1/2)^3...+(2n-1)*(1/2)^(n+1)
f(1/2)-1/2*f(1/2)=1/2+2*(1/2)^2+2*(1/2)^3+...+2*(1/2)^n-(2n-1)*(1/2)^(n+1)
可得1/2*f(1/2)=1/2+1/2+(1/2)^2+(1/2)^3+……+(1/2)^(n-1)-(2n-1)*(1/2)^(n+1)
即
f(1/2)=[1+(1/2)^2+(1/2)^3+……+(1/2)^(n-1)-(2n-1)*(1/2)^(n+1)]*2
=2+[(1/2)+(1/2)^2+……+(1/2)^(n-2)-(2n-1)*(1/2)^n]
而[(1/2)+(1/2)^2+……+(1/2)^(n-2)]lt;1
故f(1/2)lt;3
f(1/2)-1/2*f(1/2)=1/2+2*(1/2)^2+2*(1/2)^3+...+2*(1/2)^n-(2n-1)*(1/2)^(n+1)
可得1/2*f(1/2)=1/2+1/2+(1/2)^2+(1/2)^3+……+(1/2)^(n-1)-(2n-1)*(1/2)^(n+1)
即
f(1/2)=[1+(1/2)^2+(1/2)^3+……+(1/2)^(n-1)-(2n-1)*(1/2)^(n+1)]*2
=2+[(1/2)+(1/2)^2+……+(1/2)^(n-2)-(2n-1)*(1/2)^n]
而[(1/2)+(1/2)^2+……+(1/2)^(n-2)]lt;1
故f(1/2)lt;3
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