HMMT 二月 2010 · CALC 赛 · 第 4 题

HMMT February 2010 — CALC Round — Problem 4

解析

[ 4 ] Compute lim . n →∞ n 2 Answer: The main idea lies on the fact that positive integers are uniformly distributed modulo π π . (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 equivalent to the average value (using the Riemann summation) of the function | cos( k ) | over the interval [0 , π ]. ∫ π 1 2 Therefore, the answer is | cos( k ) | = . π 0 π