2×3分之1+3×4分之1+… +2013×2014分之1
（一）
=2*(1/3)+3*(1/4)+.+2013*(1/2014)
=2/3+3/4+.+2013/2014
=(1-1/3)+(1-1/4)+.+(1-1/2014)
=1-1/3+1-1/4+.+1-1/2014
=2012-1/3-1/4-.-1/2014
=2012-(1/3+1/4+.+1/2014)
=2012-6.6853419798.
=2005.3146580202.
（二）
=1/(2*3)+1/(3*4)+.+1/(2013*2014)
=(1/2-1/3)+(1/3-1/4)+.+(1/2013-1/2014)
=1/2-1/3+1/3-1/4+.+1/2013-1/2014
=1/2-1/2014
=1007/2014-1/2014
=1006/2014
=503/1007