1/12+1/34+1/45+...+1/20042005+1/2005*2006,

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1/1*2+1/3*4+1/4*5+...+1/2004*2005+1/2005*2006,
1/1*2+1/3*4+1/4*5+...+1/2004*2005+1/2005*2006,

1/1*2+1/3*4+1/4*5+...+1/2004*2005+1/2005*2006,
解
原式
=(1-1/2)+(1/2-1/3)+……+(1/2005-1/2006)
=1+(1/2-1/2)+(1/3-1/3)+……+(1/2005-1/2005)-1/2006
=1-1/2006
=2005/2006
裂项
1/n(n+1)=1/n-1/(n+1)