Why is 1/(n+1)! times a series less than 1/(n!n)?

[Unassuming]

Why is the first part of this inequality true?

1/(n+1)! [ (1 +1/(n+1) +1/(n+1)^{2} +...+ 1/(n+1)^{k} ]
< 1/(n!n) < 1/n

 

[HallsofIvy]

Let a= 1/(n+1) and that sum becomes 1+ a+ a^2+ ...+ a^k, a geometric series. You can write down a simple for for it. Once you have simplified that, it should be clear.