1/1*2+1/2*3+1/3*4+1/4*5...+1/2002*2003+1/2003*2004+1/2004*2005

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1/1*2+1/2*3+1/3*4+1/4*5...+1/2002*2003+1/2003*2004+1/2004*2005

1/1*2+1/2*3+1/3*4+1/4*5...+1/2002*2003+1/2003*2004+1/2004*2005
1/1*2+1/2*3+1/3*4+1/4*5...+1/2002*2003+1/2003*2004+1/2004*2005

1/1*2+1/2*3+1/3*4+1/4*5...+1/2002*2003+1/2003*2004+1/2004*2005
1/n(n+1)=1/n-1/(n+1)
1/1*2+1/2*3+1/3*4+1/4*5...+1/2002*2003+1/2003*2004+1/2004*2005
=1/1-1/2+1/2-1/3+1/3-1/4+……+1/2003-1/2004+1/2004-1/2005
=1-1/2005
=2004/2005