|1/3-1/2|+|1/4-1/3|+|1/5-1/4|+...+|1/2011-1/2010|=|1/2-1/3|+|1/3-1/4|+|1/4-1/5|+...+|1/2010-1/2011|=1/2-1/3+1/3-1/4+1/4-1/5+...+1/2010-1/2011= 1/2 - 1/2011=(2011-2)/4022=2009/4022