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HMMT 二月 2008 · CALC 赛 · 第 6 题

HMMT February 2008 — CALC Round — Problem 6

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 5 ] Determine the value of lim . n →∞ k k =0 ( ) √ √ √ 3 3 p 3
解析
  1. [ 5 ] Determine the value of lim . n →∞ k k =0 1 ( ) ( ) − 1 − 1 n n Answer: 2 Let S denote the sum in the limit. For n ≥ 1, we have S ≥ + = 2. On n n 0 n the 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) n 0 1 n − 1 n k n 2 k =2 which goes to 2 as n → ∞ . Therefore, S → 2. n ( √ ) √ √ 3 3 p 3