1/(12)+1/(23)+1/(34)+.1/(20072008)=

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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*2)+1/(2*3)+1/(3*4

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*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