Ta có:
\( \displaystyle{1 \over {\sqrt 2 + \sqrt 1 }} + {1 \over {\sqrt 3 + \sqrt 2 }} + {1 \over {\sqrt 4 + \sqrt 3 }}\)
\( \displaystyle = {{\sqrt 2 - \sqrt 1 } \over {(\sqrt 2 + \sqrt 1 )(\sqrt 2 - \sqrt 1 )}} \)\(\displaystyle + {{\sqrt 3 - \sqrt 2 } \over {(\sqrt 3 + \sqrt {2)} (\sqrt 3 - \sqrt 2 )}} \)\(\displaystyle + {{\sqrt 4 - \sqrt 3 } \over {(\sqrt 4 + \sqrt 3 )(\sqrt 4 - \sqrt 3 )}}\)
\( \displaystyle = {{\sqrt 2 - \sqrt 1 } \over {2 - 1}} \)\(\displaystyle + {{\sqrt 3 - \sqrt 2 } \over {3 - 2}} \)\(\displaystyle + {{\sqrt 4 - \sqrt 3 } \over {4 - 3}}\)
\( \displaystyle = \sqrt 2 - \sqrt 1 + \sqrt 3 - \sqrt 2 \)\( + \sqrt 4 - \sqrt 3 \)
\( \displaystyle = - \sqrt 1 + \sqrt 4 \)\( = - 1 + 2 = 1\)
