1/(1*2)+1/(2*3)+1/(3*4)+.1/(2007*2008)=

1/(1*2)+1/(2*3)+1/(3*4)+。1/(2007*2008)= (1/1-1/2)+(1/2-1/3)+(1/3-1/4)+。+(1/2007-1/2008)= 1-1/2008 【中间的项正好正负抵消,只剩下头尾两项】= 2007/2008