How to simplify $\frac{n!}{n^n}(n+1)^n+\frac{1}{(n+1)!}$

Question

$$\frac{n!}{n^n}(n+1)^n+\frac{1}{(n+1)!}$$

first I wrote $n!$ as $n(n+1)!$ and cancelled out the $(n+1)!$ with the $(n+1)!$ in the denominator which gave me:

$$\frac{n}{n^n}(n+1)^n+1$$

then I tried writing it like:

$$\frac{n}{n^n}(n+1)^n(n+1)$$

then I am stuck.

answer says its $\left( 1+\dfrac{1}{n} \right)^n$ but I am not sure how to attain it.

sorry for the bad editing I am on phone app.

Answer 1

$n! = n(n-1)!$

Here is a simple way to do it \begin{align*} \frac{n!}{n^n}\frac{(n+1)^{n+1}}{(n+1)!} &= \frac{1}{n^n}\frac{(n+1)^{n+1}}{n+1} \\ &= \frac{1}{n^n}(n+1)^n \\ &= \left(\frac{n+1}{n}\right)^n \\ &= \left(1+\frac{1}{n}\right)^n \end{align*}