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HMMT 二月 2008 · 冲刺赛 · 第 14 题

HMMT February 2008 — Guts Round — Problem 14

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 8 ] Evaluate the infinite sum . 4 n =1 n +4
解析
  1. [ 8 ] Evaluate the infinite sum . 4 n =1 n +4 3 Answer: We have 8 ∞ ∞ ∑ ∑ n n = 4 2 2 n + 4 ( n + 2 n + 2)( n − 2 n + 2) n =1 n =1 ( ) ∞ ∑ 1 1 1 = − 2 2 4 n − 2 n + 2 n + 2 n + 2 n =1 ( ) ∞ ∑ 1 1 1 = − . 2 2 4 ( n − 1) + 1 ( n + 1) + 1 n =1 ( ) 1 1 1 3 Observe that the sum telescopes. From this we find that the answer is + = . 2 2 4 0 +1 1 +1 8