HMMT 二月 2005 · 冲刺赛 · 第 22 题

英文解析

22. Find
    2 4 8
    { ln(1 + e ) } + { ln(1 + e ) } + { ln(1 + e ) } + { ln(1 + e ) } + · · · ,
    where { x } = x − b x c denotes the fractional part of x .
    Solution: 1 − ln( e − 1)
    k k k
    2 k 2 2
    Since ln(1 + e ) is just larger than 2 , its fractional part is ln(1 + e ) − ln e =
    − 2 kln(1 + e ). But now notice that
    ∏nk n +1
    2 2 2 − 1
    (1 + x ) = 1 + x + x + · · · + x .
    k =0
    (This is easily proven by induction or by noting that every nonnegative integer lessn +1
    than 2 has a unique ( n +1)-bit binary expansion.) If | x | < 1, this product convergesto as n goes to infinity. Therefore,1
    1 − x
    ∞ ∞
    ∑ ∏
    1 ek k − 2 − 1 2
    ln(1 + e ) = ln (1 + ( e ) ) = ln = ln = 1 − ln( e − 1) .
    − 1
    1 − e e − 1
    k =0 k =0