(1+1/2+1/3+…+1/2002)*(1/2+1/3+1/4+…+1/2003)-(1/2+1/3+1/4+…+1/2002)*(1+1/2+1/3+…+1/2003)
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![(1+1/2+1/3+…+1/2002)*(1/2+1/3+1/4+…+1/2003)-(1/2+1/3+1/4+…+1/2002)*(1+1/2+1/3+…+1/2003)](/uploads/image/z/634660-52-0.jpg?t=%EF%BC%881%2B1%2F2%2B1%2F3%2B%E2%80%A6%2B1%2F2002%EF%BC%89%2A%EF%BC%881%2F2%2B1%2F3%2B1%2F4%2B%E2%80%A6%2B1%2F2003%EF%BC%89-%281%2F2%2B1%2F3%2B1%2F4%2B%E2%80%A6%2B1%2F2002%29%2A%EF%BC%881%2B1%2F2%2B1%2F3%2B%E2%80%A6%2B1%2F2003%EF%BC%89)
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(1+1/2+1/3+…+1/2002)*(1/2+1/3+1/4+…+1/2003)-(1/2+1/3+1/4+…+1/2002)*(1+1/2+1/3+…+1/2003)
(1+1/2+1/3+…+1/2002)*(1/2+1/3+1/4+…+1/2003)-(1/2+1/3+1/4+…+1/2002)*(1+1/2+1/3+…+1/2003)
(1+1/2+1/3+…+1/2002)*(1/2+1/3+1/4+…+1/2003)-(1/2+1/3+1/4+…+1/2002)*(1+1/2+1/3+…+1/2003)
令a=1/2+1/3+…+1/2002
则原式=(1+a)(a+1/2003)-a(1+a+1/2003)
=(1+a)a+1/2003(1+a)-(1+a)a-1/2003a
=1/2003(1+a)-1/2003a
=1/2003+1/2003a-1/2003a
=1/2003
设m=1/2+1/3+。。。+1/2002
原式=(1+m)*(m+1/2003)-m*(1+m+1/2003)
=m+m*m+1/2003+1/2003*m-m-m*m-1/2003*m
=1/2003
令1/2+1/3+…+1/2002为A
则(1+A)*(A+1/2003)-A*(1+A+1/2003)= 1/2003
设m=1/2+1/3+...+1/2002
则原式
=(1+m)(m+1/2003)-m(1+m+1/2003)
=m^2+m/2003+m+1/2003-(m+m^2+1/2003*m)
=m^2+m/2003+m+1/2003-m-m^2-m/2003
=1/2003