HMMT 二月 2006 · CALC 赛 · 第 4 题

英文解析

4. Compute .
   k !
   k =1
   Answer: 15 e
   Solution: Define, for non-negative integers n ,
   ∞
   ∑nk
   S := ,
   k !nk =0
   where 0 = 1 when it occurs. Then S = e , and, for n ≥ 1,01
   ( )0
   ∞ ∞ ∞ ∞ n − 1
   n n n n − 1
   ∑ ∑ ∑ ∑ ∑
   k k ( k + 1) ( k + 1) n − 1
   S = = = = = S ,
   n ik ! k ! ( k + 1)! k ! ii =0
   k =0 k =1 k =0 k =0
   so we can compute inductively that S = e , S = 2 e , S = 5 e , and S = 15 e .
   1 2 3 4