Question

如和求证(1/n+1)*(1+1/3+1/5+…+1/2n-1)>(1/n)*(1/2+1/4+…1/2n)

Answer 1

证明：

当k=1时

1/2+1/3+1/4=13/12=26/24>25/24

结论成立.

假设k=n时结论成立,即

1/(n+1)+1/(n+2)+1/(n+3)+...+1/(3n+1)>25/24

当k=n+1时

由于

9(n+1)^2=9n^2+18n+9>9n^2+18n+8=(3n+2)(3n+4)

即

9(n+1)^2/[(3n+2)(3n+4)]-1>0

左侧为

1/[(n+1)+1]+1/[(n+1)+2]+1/[(n+1)+3]+...+1/[3(n+1)+1]

=1/(n+1)+1/(n+2)+1/(n+3)+...+1/(3n+1)+{1/(3n+2)+1/(3n+3)+1/(3n+4)-1/(n+1)}

=1/(n+1)+1/(n+2)+1/(n+3)+...+1/(3n+1)+{6(n+1)/[(3n+2)(3n+4)]-2/(3n+3)}

=1/(n+1)+1/(n+2)+1/(n+3)+...+1/(3n+1)+2/(3n+3)*{9(n+1)^2/[(3n+2)(3n+4)]-1}

>1/(n+1)+1/(n+2)+1/(n+3)+...+1/(3n+1)>25/24.

结论成立.