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HMMT 十一月 2014 · 冲刺赛 · 第 22 题

HMMT November 2014 — Guts Round — Problem 22

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 12 ] Evaluate the infinite sum
    ( )
    ∞
    ∑1
    1 −
    log .n
    1 −12
    n +1
    n =2
解析

英文解析

  1. [ 12 ] Evaluate the infinite sum
    ( )
    ∞
    ∑1
    1 −
    log .n
    1 −12
    n +1
    n =2
    ( )
    Answer: − 1 Using the identity log = log a − log b , the sum becomesa
    2 2 2
    ( ) ( )b
    ∞ ∞
    ∑ ∑
    n − 1 nlog − log .
    2 2
    n n + 1
    n =2 n =2
    ( )
    Most of the terms cancel out, except the log term from the first sum. Therefore, the answer is 1 − 1.22