1.设f(x)=(x^n-x^(-n))/(x^n+x^(-n)),n为正整数,试比较f(根号2)与(n^2-1)/(n^2+1)的大小,并说明理由.2.已知各项都为正数的数列an满足an小于等于根号an-根号a(n+1),求证,当n大于等于2时,an小于等于1/(n+2)^2

Question

1.设f(x)=(x^n-x^(-n))/(x^n+x^(-n)),n为正整数,试比较f(根号2)与(n^2-1)/(n^2+1)的大小,并说明理由.
2.已知各项都为正数的数列an满足an小于等于根号an-根号a(n+1),求证,当n大于等于2时,an小于等于1/(n+2)^2

Answer 1

1.
f(x)=(x^n-x^(-n))/(x^n+x^(-n))=(x^2n-1)/(x^2n+1)=1-2/(x^2n+1)
(n^2-1)/(n^2+1)=1-2/(n^2+1)
f(2^(1/2))=1-2/2^n+1
n=1时 2^n>n^2 f(2^(1/2))>(n^2-1)/(n^2+1)
n=2时 2^n=n^2=4 f(2^(1/2))=(n^2-1)/(n^2+1)
n=3时 2^n=8n^2 f(2^(1/2))>(n^2-1)/(n^2+1)
2、 用数学归纳法证明bn=(an)^(1/2)