证明:对任意的正整数n,有1/1×3+1/2×4+1/3×5+.+1/n(n+2)

问题描述:

证明:对任意的正整数n,有1/1×3+1/2×4+1/3×5+.+1/n(n+2)

设an= 1/n(n+2) = 1/2* [1/n - 1/(n+2)] (n>=1)
设an的和为Sn
则Sn =1/2* [1/1-1/3+1/2-1/4+1/3-1/5……+1/(n-1)-1(n+1)+1/n-1/(n+2)] 仔细观察 这里面各项抵消
=1/2 *[1+1/2-1/(n+1)-1(n+2)]
很明显 1/(n+1) 和 1/(n+2)始终大于0
有Sn得证

原式=1/2[1-1/3+1/2-1/4+1/3-1/5+.+1/n-1/(n+2)]
=1/2[1+1/2-1/n-1/(n+2)]
=3/4-1/n-1/(n+2)