Understanding the First Part of an Inequality Factorial

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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
 
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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.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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