HMMT 二月 2006 · 冲刺赛 · 第 40 题

英文解析

40. Compute
    ∞
    ∑
    3 k + 1
    k +1
    · ( − 1) .
    3 2
    2 k + kk =1
    π π2
    Answer: + − 2 + ln 2
    12 2
    Solution: Via partial fraction decomposition we write the sum as
    ( )
    ∞
    ∑
    1 2 1
    k +1 − + ( − 1)
    k 1 + 2 k k 2
    k =1
    Now recall that
    ∞
    ∑2
    1 π = = Sk 621
    k =1
    ∞
    k +1
    ∑
    ( − 1) = ln(2) = Sk 2
    k =1
    ∞
    k +1
    ∑
    ( − 1) π = = S
    2 k − 1 43
    k =1
    Manipulating (1), we deduce 14
    ( )
    ∞ ∞ ∞
    k +1
    ∑ ∑ ∑
    ( − 1) 1 1 = − 2 ·
    2 2 2
    k k (2 k )
    k =1 k =1 k =1
    2 2 2
    π π π = − 2 / 4 · = = S
    6 6 124
    It is then easily seen that the answer is equal to S + 2 · S − 2 + S .
    2 3 4