1/(1+√2)=(√2-1)/(√2-1)(√2+1)=(√2-1)/(2-1)=√2-1
同理,1/(√2+√3)=(√3-√2)/(3-2)=√3-√2
后面的以此类推
所以
1/(1+√2)+1/（√2＋√3）+.1/(√99+√100)
=(√2-1)+(√3-√2)+……+(√100-√99)
=√2-1+√3-√2+……+√100-√99
=√100-1