Sen asked in 科學數學 · 1 decade ago

重要的數學證明

試證明

1 + 1/2 + 1/3 + 1/4........................+ 1/n 之和不為整數

Update:

麻煩說更清些

3 Answers

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  • 1 decade ago
    Favorite Answer

    consider H_n, n>1 choose k such that 2^k <= n <= 2^(k+1)

    For example, if n=12, then k=3.

    H_n=1+1/2+1/3+...+1/n

    Let M be the LCM of all the denominators except 2^k

    M=LCM(1,2,3,...,2^k-1,2^k+1,...n)

    A crucial point is that M has a factor 2^(k-1) but not 2^k

    Multiply H_n by M,

    M*H_n=M+M/2+M/3+...+M/2^k+...+M/n

    =integer+M/2^k+integer

    Based on our definition of M, M/2^k cannot be an integer. Therefore M*H_n cannot be an integer, and so H_n cannot be an integer as well.

  • 1 decade ago

    柚子老兄您答非所問= =

    看起來很簡單 不過算一算發現好難= =

  • 1 decade ago

    1/2也就是一半,

    1/3也就是3/2

    但是1/4+1/2=3/4

    也就是說,1/2+1/4+1/8+...(都是能夠自己通分的)...+1/2n<1

    SO...不管怎麼家都不會變1,

    也就是說其他的數字也是。

    你看的懂嗎...?

    看不懂在跟我說一下,我換個方式說。

    2010-01-21 14:27:47 補充:

    恩...

    1/2+1/4+1/8+...(都是能夠自己通分的)...+1/2n<1

    這你看的懂嗎?

    因為這是解題關鍵。

    假設n=8

    1/1+1/2+1/3+1/4+1/5+1/6+1/7+1/8

    求1~8的最大公因數為840

    =(840 + 420 + 280 + 210 + 168 + 140 + 120 + 105) ÷ 840

    1 2 3 4 5 6 7 8

    =2283÷840不是整數

    所以,不管N多大多小,都不是整數。除了N=1之外。

    這樣看的懂嗎?

    2010-02-12 17:43:13 補充:

    有嗎...

    我覺得這樣想OK阿...

    不行嗎...?

    Source(s):
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