First of all, we can judge that the value in the absolute value sign must be negative, so we must take the inverse number to remove the absolute value sign.

Namely:-(1/2010-1/2009)-(1/2009-1/2008) ......

Then remove the brackets:

- 1/20 10+ 1/2009- 1/2009+ 1/2008- 1/2008...- 1/3+ 1/2- 1/2+ 1

Merge and simplify:

- 1/20 10+ 1=2009/20 10

-1/20 10 and 1, the middle numbers plus one minus one plus one minus one all cancel out. Understand?