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Constructing polynomials and proving limit

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October 12, 2010
algebrapolynomiallimitalgebra unsolved

Problem Statement

Let f1(x)=x3+a1x2+b1x+c1=0f_1(x) = x^3+a_1x^2+b_1x+c_1 = 0 be an equation with three positive roots α>β>γ>0\alpha>\beta>\gamma > 0. From the equation f1(x)=0f_1(x) = 0, one constructs the equation f2(x)=x3+a2x2+b2x+c2=x(x+b1)2−(a1x+c1)2=0f_2(x) = x^3 +a_2x^2 +b_2x+c_2 = x(x+b_1)^2 -(a_1x+c_1)^2 = 0. Continuing this process, we get equations f3,⋯ ,fnf_3,\cdots, f_n. Prove that lim⁡n→∞−an2n−1=α\lim_{n\to\infty}\sqrt[2^{n-1}]{-a_n} = \alpha
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