数列{an} 定义如下,a1=2,an+1=an^2-an+1,求证1/a1+1/a2+1/a3+……+1/a2008
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数列{an} 定义如下,a1=2,an+1=an^2-an+1,求证1/a1+1/a2+1/a3+……+1/a2008
数列{an} 定义如下,a1=2,an+1=an^2-an+1,求证1/a1+1/a2+1/a3+……+1/a2008<1
数列{an} 定义如下,a1=2,an+1=an^2-an+1,求证1/a1+1/a2+1/a3+……+1/a2008
符号说明:a[n]中n为下标.
由题意,a[n+1]-1=a[n]*(a[n]-1)
等式两边取倒数,得到1/(a[n+1]-1)=1/(a[n]-1)-1/a[n]
于是有1/a[n]=1/(a[n]-1)-1/(a[n+1]-1),对任意正整数n成立
于是
1/a[1]+1/a[2]+1/a[3]+……+1/a[2008]
=1/(a[1]-1)-1/(a[2009]-1)
=1-1/(a[2009]-1)
要证此式小于1,只需再证a[2009]>1
而a[n+1]-a[n]=a[n]^2-2*a[n]-1=(a[n]-1)^2≥0
因此对任意n>1有a[n]≥a[1]=2,所以a[2009]≥2>1成立,故命题得证.