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HMMT 二月 2008 · 冲刺赛 · 第 14 题

HMMT February 2008 — Guts Round — Problem 14

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 8 ] Evaluate the infinite sum .
    n =14
    n +4
解析

英文解析

  1. [ 8 ] Evaluate the infinite sum .
    n =14
    n +4
    Answer: We have 3
    ∞ ∞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