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HMMT 十一月 2014 · 冲刺赛 · 第 22 题

HMMT November 2014 — Guts Round — Problem 22

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 12 ] Evaluate the infinite sum ( ) ∞ 1 ∑ 1 − n log . 2 1 1 − n +1 n =2
解析
  1. [ 12 ] Evaluate the infinite sum ( ) ∞ 1 ∑ 1 − n log . 2 1 1 − n +1 n =2 ( ) a Answer: − 1 Using the identity log = log a − log b , the sum becomes 2 2 2 b ( ) ( ) ∞ ∞ ∑ ∑ n − 1 n log − log . 2 2 n n + 1 n =2 n =2 ( ) 1 Most of the terms cancel out, except the log term from the first sum. Therefore, the answer is 2 2 − 1.