设1/2+1/3+...+1/1997=a,1/2+1/3+...+1/1996=b
原式=a(1+b)-(1+a)b
=a+ab-b-ab
=a-b
=1/2+1/3+...+1/1997-1/2-1/3+...-1/1996
=1/1997