1/12+1/34+1/45+.+1/20082009

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1/1*2+1/3*4+1/4*5+.+1/2008*20091/1*2+1/3*4+1/4*5+.+1/2008*20091/1*2+1/3*4+1/4*5+.+1/2008*2009裂项少一个1/

1/1*2+1/3*4+1/4*5+.+1/2008*2009
1/1*2+1/3*4+1/4*5+.+1/2008*2009

1/1*2+1/3*4+1/4*5+.+1/2008*2009
裂项
少一个1/2*3吧?
1/1*2+1/2*3+1/3*4+1/4*5+.+1/2008*2009
=1-1/2+1/2-1/3+...+1/2008-1/2009
=1-1/2009
=2008/2009

简便算法：1/2*1/3=1/6=1/2-1/3
1/3*1/4=1/12=1/3-1/4
以此类推，则1/2008*1/2009=1/2008-1/2009
所以原式=1/2-1/3+1/3-1/4+1/4-1/5……+1/2008-1/2009
=1/2-1/2009
=2007/4018

1/1*2+1/2*3+1/3*4+1/4*5+.....+1/2008*2009
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+.......................+1/2008-1/2009
=1-1/2009
=2008/2009