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已知 a(n+1) = 2a(n)^2 / (a(n)^2+1),所以 1/(a(n)+1) = (a(n)^2+1)/(3a(n)^2+1) = 1/3 + 2/ (3 * (3a(n)^2+1))
又由第一问得知 a(n) ≤ 1/(2n+1)
带入上式化简,得到 1/(a(n)+1) = 1 - 2 / ((2n+1)^2+3) = 1 - 1 / (2n^2+2n+1) > 1 - 1 / (2n^2+2n) = 1 - 1/2 * (1/n - 1/(n+1))
所以1/(a(1)+1)+1/(a(2)+1)+... +1/(a(n)+1) > n - 1/2 *(1- 1/(n+1)) > n - 3/4
又由第一问得知 a(n) ≤ 1/(2n+1)
带入上式化简,得到 1/(a(n)+1) = 1 - 2 / ((2n+1)^2+3) = 1 - 1 / (2n^2+2n+1) > 1 - 1 / (2n^2+2n) = 1 - 1/2 * (1/n - 1/(n+1))
所以1/(a(1)+1)+1/(a(2)+1)+... +1/(a(n)+1) > n - 1/2 *(1- 1/(n+1)) > n - 3/4
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