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36

Part of 2005 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 36

Source: 2004 Nagoya University

6/6/2005
A sequence of polynomial fn(x) (n=0,1,2,⋯ )f_n(x)\ (n=0,1,2,\cdots) satisfies f0(x)=2,f1(x)=xf_0(x)=2,f_1(x)=x, fn(x)=xfn−1(x)−fn−2(x), (n=2,3,4,⋯ )f_n(x)=xf_{n-1}(x)-f_{n-2}(x),\ (n=2,3,4,\cdots) Let xn (n≧2)x_n\ (n\geqq 2) be the maximum real root of the equation fn(x)=0 (∣x∣≦2)f_n(x)=0\ (|x|\leqq 2) Evaluate lim⁡n→∞n2∫xn2fn(x)dx\lim_{n\to\infty} n^2 \int_{x_n}^2 f_n(x)dx
calculusintegrationalgebrapolynomiallimitcalculus computations