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HMMT 二月 2001 · CALC 赛 · 第 8 题

HMMT February 2001 — CALC Round — Problem 8

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Evaluate cot ( n + n + 1). n =0
解析
  1. Evaluate cot ( n + n + 1). n =0 ∞ ∞ ∞ ∑ ∑ ∑ 1 2 Solution: cot( n + n + 1) = arctan( ) = arctan( n + 1) − arctan( n ) 2 n + n +1 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 .