MathDB
2009 Calculus #6: Class of Polynomials

Source:

June 23, 2012
calculusalgebrapolynomialfunction

Problem Statement

Let p0(x),p1(x),p2(x),…p_0(x),p_1(x),p_2(x),\ldots be polynomials such that p0(x)=xp_0(x)=x and for all positive integers nn, ddxpn(x)=pn−1(x)\dfrac{d}{dx}p_n(x)=p_{n-1}(x). Define the function p(x):[0,∞)→Rp(x):[0,\infty)\to\mathbb{R} by p(x)=pn(x)p(x)=p_n(x) for all x∈[n,n+1)x\in [n,n+1). Given that p(x)p(x) is continuous on [0,∞)[0,\infty), compute ∑n=0∞pn(2009).\sum_{n=0}^\infty p_n(2009).
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