First we must see that for the first term :
( 1- ( 1/2 )^2 is in the form of a^2 - b^2 where a^2 = 1^2 and b^2 = ( 1/2 )^2
Then we change all the terms into the form of a^2 - b^2 = ( a-b )( a+b )
= [ 1 - ( 1/2 )^2 ] x [ 1 - ( 1/3 )^2 ] x [ 1 - ( 1/4 )^2 ] ... ... [ 1 - ( 1/2007 )^2 ]
= [( 1 - 1/2 )( 1 + 1/2 )] x [( 1 - 1/3 )( 1 + 1/3 )] x [( 1 - 1/4 )( 1 + 1/4 )] ... [( 1 - 1/2007)( 1 + 1/2007 )]
= 1/2 x 3/2 x 2/3 x 4/3 x 3/4 x 5/4 ..... 2006/2007 x 2008/2007
= 1/2 x 2008/2007 [ The rest of the terms cancels each other out , leaving 1/2 and 2008/2007 ]
= 1004 / 2007
