HMMT 二月 2009 · CALC 赛 · 第 6 题
HMMT February 2009 — CALC Round — Problem 6
题目详情
英文原题
- [ 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
p ( x ) = p ( x ). Define the function p ( x ) : [0 , ∞ ) → R by p ( x ) = p ( x ) for all x ∈ [ n, n + 1). Givendn n − 1 nthat p ( x ) is continuous on [0 , ∞ ), computedx
∞
∑
p (2009) .
n =0 n
解析
英文解析
- [ 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
p ( x ) = p ( x ). Define the function p ( x ) : [0 , ∞ ) → R x by p ( x ) = p ( x ) for all x ∈ [ n, n + 1].dn n − 1 n
Given that p ( x ) is continuous on [0 , ∞ ), computedx
∞
∑
p (2009) .
n =0 n
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 expansionn
( n +1)! n !
of e .x