HMMT 二月 2010 · CALC 赛 · 第 4 题

英文解析

4. [ 4 ] Compute lim .
   n →∞
   2 n
   Answer: The main idea lies on the fact that positive integers are uniformly distributed moduloππ . (In the other words, if each integer n is written as qπ + r where q is an integer and 0 ≤ r < π , the value of r will distribute uniformly in the interval [0 , π ].) Using this fact, the summation is equivalentto the average value (using the Riemann summation) of the function | cos( k ) | over the interval [0 , π ].
   ∫
   1 2π
   Therefore, the answer is | cos( k ) | = .
   π 0 π