(1/2+1/3...+1/2003)(1+1/2+1/3+...+1/2002)-(1+1/2+1/3+...+1/2003)(1/2+1/3+...+1/2002)

问题描述:

(1/2+1/3...+1/2003)(1+1/2+1/3+...+1/2002)-(1+1/2+1/3+...+1/2003)(1/2+1/3+...+1/2002)

设A=1/2+1/3...+1/2003,B=1/2+1/3+...+1/2002,所以,
(1/2+1/3...+1/2003)(1+1/2+1/3+...+1/2002)-(1+1/2+1/3+...+1/2003)(1/2+1/3+...+1/2002)
=A*(1+B)-(1+A)*B
=A+A*B-B-A*B
=A-B
=1/2+1/3...+1/2003-1/2-1/3-...-1/2002
=1/2003