(1/2+1/3+...+1/2004)(1+1/2+1/3+...+1/2003)-(1+1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)
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![(1/2+1/3+...+1/2004)(1+1/2+1/3+...+1/2003)-(1+1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)](/uploads/image/z/14447620-28-0.jpg?t=%281%2F2%2B1%2F3%2B...%2B1%2F2004%29%EF%BC%881%2B1%2F2%2B1%2F3%2B...%2B1%2F2003%EF%BC%89-%281%2B1%2F2%2B1%2F3%2B...%2B1%2F2004%EF%BC%89%281%2F2%2B1%2F3%2B...%2B1%2F2003%29)
(1/2+1/3+...+1/2004)(1+1/2+1/3+...+1/2003)-(1+1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)
(1/2+1/3+...+1/2004)(1+1/2+1/3+...+1/2003)-(1+1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)
(1/2+1/3+...+1/2004)(1+1/2+1/3+...+1/2003)-(1+1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)
(1/2+1/3+...+1/2004)(1+1/2+1/3+...+1/2003)-(1+1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)
=(1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)+(1/2+1/3+...+1/2004)-(1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)-(1/2+1/3+...+1/2003)
=(1/2+1/3+...+1/2004)-(1/2+1/3+...+1/2003)
=1/2004
(1/2+1/3+...+1/2004)(1+1/2+1/3+...+1/2003)-(1+1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)
=(1/2+1/3+...+1/2004)+(1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)-(1/2+1/3+...+1/2003)-(1/2+1/3+...+1/2004)(1/2+1/3+...+1/2003)
=1/2004
http://zhidao.baidu.com/question/120425165.html
方法一样。
令t=1+1/2+1/3+...+1/2004
原式=(t-1)(t-1/2004)-t(t-1-1/2004)
=1/2004