1/1*2+1/2*3+1/3*4+...+1/1993*1994=

1/1*2+1/2*3+1/3*4+...+1/1993*1994=
1/1*2+1/2*3+1/3*4+...+1/1993*1994=

1/1*2+1/2*3+1/3*4+...+1/1993*1994=
1/n(n+1)=1/n-1/(n+1)
所以1/1*2+1/2*3+1/3*4+...+1/1993*1994
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/1993-1/1994)
=1-1/1994
=1993/1994
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=1/1-1/2+1/2-1/3+...+1/1992-1/1993+1/1993-1/1994
=1=1/1994=1993/1994

1/(1*2)+1/(2*3)+1/(3*4)+...+1/(1993*1994)
=(2-1)/(1*2)+(3-2)/(2*3)+(4-3)/(3*4)+...+(1994-1993)/(1993*1994)
=(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+...+(1/1993-1/1994)
=1/1-1/2+1/2-1/3+1/3-1/4+...+1/1992-1/1993+1/1993-1/1994
=1-1/1994
=1993/1994

=（1-1/2）+（1/2-1/3）+（1/3-1/4）+.....+（1/1993-1/1994）
=1-1/1994
=1993/1994

因为1/1*2=1-1/2;1/2*3=1/2-1/3;1/3*4=1/3-1/4;...1/1993*1994=1/1993-1/1994
所以1/1*2+1/2*3+1/3*4+...+1/1993*1994
=1-1/2+1/2-1/3+1/3-1/4+……+1/1993-1/1994
=1+(-1/2+1/2)+(-1/3+1/3)+(-1/4+……+1/1993)-1/1994
=1-1/1994
=1993/1994