数列竞赛题!在线等!数列{an},a1=2/3,a(n+1)=an^2+a(n-1)^2+.+a1^2(n∈N+),若对于任意n属于N+,1/(a1+1)+2/(a2+1)+.+1/(an +1)<M恒成立,求M最小值
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![数列竞赛题!在线等!数列{an},a1=2/3,a(n+1)=an^2+a(n-1)^2+.+a1^2(n∈N+),若对于任意n属于N+,1/(a1+1)+2/(a2+1)+.+1/(an +1)<M恒成立,求M最小值](/uploads/image/z/693422-62-2.jpg?t=%E6%95%B0%E5%88%97%E7%AB%9E%E8%B5%9B%E9%A2%98%21%E5%9C%A8%E7%BA%BF%E7%AD%89%21%E6%95%B0%E5%88%97%7Ban%7D%2Ca1%3D2%2F3%2Ca%28n%2B1%29%3Dan%5E2%2Ba%28n-1%29%5E2%2B.%2Ba1%5E2%28n%E2%88%88N%2B%29%2C%E8%8B%A5%E5%AF%B9%E4%BA%8E%E4%BB%BB%E6%84%8Fn%E5%B1%9E%E4%BA%8EN%2B%2C1%2F%EF%BC%88a1%2B1%EF%BC%89%2B2%2F%EF%BC%88a2%2B1%EF%BC%89%2B.%2B1%2F%28an+++%2B1%29%EF%BC%9CM%E6%81%92%E6%88%90%E7%AB%8B%2C%E6%B1%82M%E6%9C%80%E5%B0%8F%E5%80%BC)
数列竞赛题!在线等!数列{an},a1=2/3,a(n+1)=an^2+a(n-1)^2+.+a1^2(n∈N+),若对于任意n属于N+,1/(a1+1)+2/(a2+1)+.+1/(an +1)<M恒成立,求M最小值
数列竞赛题!在线等!
数列{an},a1=2/3,a(n+1)=an^2+a(n-1)^2+.+a1^2(n∈N+),若对于任意n属于N+,1/(a1+1)+2/(a2+1)+.+1/(an +1)<M恒成立,求M最小值
数列竞赛题!在线等!数列{an},a1=2/3,a(n+1)=an^2+a(n-1)^2+.+a1^2(n∈N+),若对于任意n属于N+,1/(a1+1)+2/(a2+1)+.+1/(an +1)<M恒成立,求M最小值
题目抄错了吧,应该是1/(a2+1)吧.
n≥2时,
a(n+1)=an²+a(n-1)²+...+a1² (1)
an=a(n-1)²+a(n-2)²+...+a1² (2)
(1)-(2)
a(n+1)-an=an²
a(n+1)=an²+an=an(an +1)
1/a(n+1)=1/[an(an +1)]=1/an -1/(an +1)
1/(an +1)=1/an- 1/a(n+1)
1/(a1+1)+1/(a2+1)+...+1/(an +1)
=1/a1-1/a2+1/a2-1/a3+...+1/an -1/a(n+1)
=1/a1 -1/a(n+1)
a(n+1)/an=an(an +1)/an=an +1>1,数列是递增数列.
n->+∞时,a(n+1)->+∞ 1/a(n+1)->0
1/a1 -1/a(n+1)->1/a1=1/(2/3)=3/2
1/a1 -1/a(n+1)