1/(√2+1)+1/(√3+√2)+1/(√4+√3)+.+1/(√2007+√2008)
=(√2-1)/(√2+1)(√2-1)+(√3-√2)/(√3+√2)(√3-√2)+(√4-√3)/(√4+√3)(√4-√3)+.+(√2008-√2007)/(√2007+√2008)(√2008-√2007)
=√2-1+√3-√2+√4-√3+.+√2008-√2007
=√2008-1
=2√502-1