a1=2,an+an-1=(n/(an-an-1))+2,求1/(a1-1)^2+1/(a2-1)^2+...+1/(an-1)^2的极限
题目
a1=2,an+an-1=(n/(an-an-1))+2,求1/(a1-1)^2+1/(a2-1)^2+...+1/(an-1)^2的极限
Thanks very much~
答案
an+an-1=(n/(an-an-1))+2 去分母整理得 (an-1)^2=(a(n-1)-1)^2+n
=>
(an-1)^2=(a(n-1)-1)^2+n
=(a(n-2)-1)^2+n-1 +n
...
= 1+2+ ...+n = n(n+1)/2
1/(an-1)^2 = 1/(n(n+1)/2) = 2/n - 2/(n+1)
==>
1/(a1-1)^2+1/(a2-1)^2+...+1/(an-1)^2 = 2 - 2/(n+1) --> 2
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