一乘二分之一加二乘三分之一加省略号加2012乘2013

1/n*（n+1）=(1/n-1/（n+1）)
1/2=1/1*2=(1/1-1/2)
1/2*3=(1/2-1/3)
1/3*4=(1/3-1/4)
.
1/2011*2012=(1/2011-1/2012)
1/2012*2013=(1/2012-1/2013)
上述各式相加得
(1/1-1/2)+(1/2-1/3)+(1/3-1/4)+.+(1/2011-1/2012)+(1/2012-1/2013)
=(1/1-1/2+1/2-1/3+1/3-1/4+.+1/2011-1/2012+1/2012-1/2013)
=(1/1-1/2013)
=2012/2013