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Putnam 1978 B3

Source: Putnam 1978

May 2, 2022
PutnamPolynomialsRecurrenceSequences

Problem Statement

The sequence (Qn(x))(Q_{n}(x)) of polynomials is defined by Q1(x)=1+x,  Q2(x)=1+2x,Q_{1}(x)=1+x ,\; Q_{2}(x)=1+2x, and for m≥1m \geq 1 by Q2m+1(x)=Q2m(x)+(m+1)xQ2m−1(x),Q_{2m+1}(x)= Q_{2m}(x) +(m+1)x Q_{2m-1}(x), Q2m+2(x)=Q2m+1(x)+(m+1)xQ2m(x).Q_{2m+2}(x)= Q_{2m+1}(x) +(m+1)x Q_{2m}(x). Let xnx_n be the largest real root of Qn(x).Q_{n}(x). Prove that (xn)(x_n ) is an increasing sequence and that lim⁡n→∞xn=0.\lim_{n\to \infty} x_n =0.
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