Question

求数列极限

数列lim[1/(1×2×3)+1/(2×3×4)+1/(3×4×5)+....+1/(n-1)(n)(n+1)要解法

Answer 1

2/(k-1)(k)(k+1)＝[1/(k-1)-1/k]-[1/k-1/(k+1)] (k∈N)

把上式看成两部分，令k＝2～n，并把各式求和，即为：
2/(1×2×3)+2/(2×3×4)+2/(3×4×5)+....+2/(n-1)(n)(n+1)
＝[1/1-1/2+1/2-1/3+……+1/(n-1)-1/n] - [1/2-1/3+1/3-1/4+1/4-1/5+……+1/n-1/(n+1)]
＝(1-1/n)-[1/2-1/(n+1)]
＝1/2+1/(n+1)-1/n
＝1/2-1/[n(n+1)]

原式＝(1/2)lim{1/2-1/[n(n+1)]}＝1/4 (n→∞)