\(\displaystyle {{S}} = {1 \over 5} + {1 \over {13}} + {1 \over {14}} + {1 \over {15}} + {1 \over {61}} \)\(\displaystyle+ {1 \over {62}} + {1 \over {63}} \)
\(\displaystyle {{S}} = {1 \over 5} + \left( {{1 \over {13}} + {1 \over {14}} + {1 \over {15}}} \right) \)\(\displaystyle+ \left( {{1 \over {61}} + {1 \over {62}} + {1 \over {63}}} \right)\) \((1)\)
Ta có :
\(\displaystyle {1 \over {13}} + {1 \over {14}} + {1 \over {15}} \)\(\displaystyle < {1 \over {12}} + {1 \over {12}} + {1 \over {12}} = {1 \over 4}\) \((2)\)
\(\displaystyle {1 \over {61}} + {1 \over {62}} + {1 \over {63}} \)\(\displaystyle< {1 \over {60}} + {1 \over {60}} + {1 \over {60}} = {1 \over {20}}\) \((3)\)
\(\displaystyle {1 \over 5} + {1 \over 4} + {1 \over {20}} = {4 \over {20}} + {5 \over {20}} + {1 \over {20}} \)\(\displaystyle= {{10} \over {20}} = {1 \over 2}\) \((4)\)
Từ \((1)\), \((2)\), \((3)\) và \((4)\) suy ra:
\(\displaystyle {\rm{S}} = {1 \over 5} + {1 \over {13}} + {1 \over {14}} + {1 \over {15}} \)\(\displaystyle + {1 \over {61}} + {1 \over {62}} + {1 \over {63}} \)\(\displaystyle < {1 \over 5} + {1 \over 4} + {1 \over {20}} = {1 \over 2}.\)
