1/1x2+1/2x3+1/3x4+1/4x5+```````+1/2008x2009=?

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1/1x2+1/2x3+1/3x4+1/4x5+```````+1/2008x2009=?1/1x2+1/2x3+1/3x4+1/4x5+```````+1/2008x2009=?1/1x2+1/2x

1/1x2+1/2x3+1/3x4+1/4x5+```````+1/2008x2009=?
1/1x2+1/2x3+1/3x4+1/4x5+```````+1/2008x2009=?

1/1x2+1/2x3+1/3x4+1/4x5+```````+1/2008x2009=?
裂项1/n*(n+1)=1/n-1/n+1
原式=(1-1/2)+(1/2-1/3)+(1/3-1/4)+.+(1/2008-1/2009)
=1-1/2009
=2008/2009

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

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