1/（1+√2）+1/（√2+√3）+1/（√3+√4）+……+1/（√8+√9）
=(1-√2)/[(1+√2)(1-√2)]+(√2-√3)/[(√2+√3)(√2-√3)]+……+（√8-√9)/[(√8+√9)(√8-√9)]
=(1-√2)/(1-2)+(√2-√3)/(2-3)+……+（√8-√9)/(8-9)
=-(1-√2)+[-(√2-√3)]+……+[-(√8-√9）]
=-1+√2-√2+√3+……-√8+√9
=-1+（√2-√2）+（√3-√3）+……+（√8-√8）+√9
=-1+0+0+……+0+√9
=-1+√9
=-1+3
=2