HMMT 二月 2009 · CALC 赛 · 第 6 题

HMMT February 2009 — CALC Round — Problem 6

[ 5 ] Let p ( x ) , p ( x ) , p ( x ) , . . . be polynomials such that p ( x ) = x and for all positive integers n , 0 1 2 0 d p ( x ) = p ( x ). Define the function p ( x ) : [0 , ∞ ) → R by p ( x ) = p ( x ) for all x ∈ [ n, n + 1). Given n n − 1 n dx that p ( x ) is continuous on [0 , ∞ ), compute ∞ ∑ p (2009) . n n =0

解析

[ 5 ] Let p ( x ) , p ( x ) , p ( x ) , . . . be polynomials such that p ( x ) = x and for all positive integers n , 0 1 2 0 d p ( x ) = p ( x ). Define the function p ( x ) : [0 , ∞ ) → R x by p ( x ) = p ( x ) for all x ∈ [ n, n + 1]. n n − 1 n dx Given that p ( x ) is continuous on [0 , ∞ ), compute ∞ ∑ p (2009) . n n =0 2010 2009 Answer: e − e − 1 Solution: By writing out the first few polynomials, one can guess and then show by induction that 1 1 n +1 n 2010 2009 p ( x ) = ( x + 1) − x . Thus the sum evaluates to e − e − 1 by the series expansion n ( n +1)! n ! x of e .