Subtract one-half of 2008, the remaining one-third, and the remaining one-fifth of the filter until the remaining one-third of 2008 is finally subtracted

Subtract one-half of 2008, the remaining one-third, and the remaining one-fifth of the filter until the remaining one-third of 2008 is finally subtracted

2008-2008*1/2-2008*1/2*1/3-...-2008*1/2*1/3*...*1/2008
=2008*1/2*1/3*...*1/2007*(1-1/2008)
=2008*1/2*1/3*...*1/2007*1/2007
=