HMMT 二月 2008 · CALC 赛 · 第 6 题

英文解析

6. [ 5 ] Determine the value of lim .
   n →∞
   k =0 k
   ( ) ( )1 − 1 − 1
   n n
   Answer: 2 Let S denote the sum in the limit. For n ≥ 1, we have S ≥ + = 2. Onn n
   0 nthe other hand, for n ≥ 3, we have
   ( ) ( ) ( ) ( ) ( ) ( )
   n − 2 − 1 − 1 − 1 − 1 − 1 − 1
   ∑
   n n n n n 2 n
   S = + + + + ≤ 2 + + ( n − 3)
   0 1 n − 1 n k n 2 nk =2
   which goes to 2 as n → ∞ . Therefore, S → 2.
   ( √ )n
   √
   √
   3 3
   p 3