`4/(2*4)+4/(4*6)+4/(6*8)+...+4/(2008*2010)`
`=4/2(2/(2*4)+2/(4*6)+...+2/(2008*2010)`
`=2(1/2-1/4+1/4-1/6+...+1/2008-1/2010)`
`=2[1/2+(-1/4+1/4)+(-1/6+1/6)+...+(-1/2008+1/2008)-1/2010]`
`=2(1/2-1/2010)`
`=2(1005/2010-1/2010)`
`=2*502/1005`
`=1004/1005`
$\text{@ThanhTruc2008}$