Chứng tỏ rằng ( 1 – 1/3 ) × ( 1 - 1/6 ) × ( 1 - 1/10 ) × ( 1 - 1/15 )...( 1 - 1/253 ) < 2/5 Chú ý: 1/3 , 1/6 , 1/10 , 1/15 , 1/283 , 2/5 là phân số

`( 1 − 1/3 ) ( 1 − 1/6 ) . . . ( 1 − 1/253 )`

`= 2/3 × 5/6 × . . . × 252/253`

`= 4/6 × 10/12 × . . . ⋅ 504/506`

`= (1 × 4)/(2 × 3) × (2 × 5)/(3 × 4) × . . . × (21 × 24)/(22 × 23)`

`= (1 × 2 × 3 × 4^2 × 5^2 ⋅ . . . ⋅ 21^2 × 22 × 23 × 24)/(2 × 3^2 × 4^2 × . . . × 22^2 × 23)`

`= (1 × 24)/(3 × 22)`

`= 24/66 <2/5`

`⇒ ( 1-1/3 )(1-1/6)(1-1/15)...(1-1/253) < 2/5`