Sequence of polynomials
Source: Vietnam NMO 1989 Problem 5
February 1, 2009
algebrapolynomialinequalities unsolvedinequalities
Problem Statement
The sequence of polynomials \left\{P_n(x)\right\}_{n\equal{}0}^{\plus{}\infty} is defined inductively by P_0(x) \equal{} 0 and P_{n\plus{}1}(x) \equal{} P_n(x)\plus{}\frac{x \minus{} P_n^2(x)}{2}. Prove that for any and any natural number it holds that 0\le\sqrt x\minus{} P_n(x)\le\frac{2}{n \plus{} 1}.