返回题库

HMMT 二月 2001 · CALC 赛 · 第 8 题

HMMT February 2001 — CALC Round — Problem 8

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Evaluate cot ( n + n + 1).
    n =0
解析

英文解析

  1. Evaluate cot ( n + n + 1).
    n =0
    ∞ ∞ ∞
    ∑ ∑ ∑
    Solution: cot( n + n + 1) = arctan( ) = arctan( n + 1) − arctan( n )21
    n + n +12
    n =0 n =0 n =0
    by the sum/difference formula for tangent. This sum, taken out to n = N , telescopes to − arctan(0) + arctan( N + 1). So as N goes to infinity, the sum goes to π/ 2 .