(1-1/2²)(1-1/3²)(1-1/4²).(1-1/1999²)(1-1/2000²)

(1-1/2²)(1-1/3²)(1-1/4²).(1-1/1999²)(1-1/2000²)
(1-1/2²)(1-1/3²)(1-1/4²).(1-1/1999²)(1-1/2000²)

(1-1/2²)(1-1/3²)(1-1/4²).(1-1/1999²)(1-1/2000²)
(1-1/2²)(1-1/3²)(1-1/4²).(1-1/1999²)(1-1/2000²)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)·····(1-1/2000)(1+1/2000)
=1/2×3/2×2/3×4/3×3/4×5/4×···×1999/2000×2001/2000
=1/2×2001/2000
=2001/4000

答案是2001/4000；把每个1改写成n^2/n^2然后分子可以成平方差,依次写下来就可以约去很多说,分子剩下2001；分母剩下2和2000