I can't find the analytical expression of the sum from this probability problem

It originates from this problem:

• We have a box with $n$ balls that are evenly mixed, all are
  $color{blue}{blue}$ except for one that is $color{red}{red}$.
• If we pick blindly one ball per try and after each try we don’t put
  it back in the box, what is the average number of tries it will take
  us to take get the $color{red}{red}$ ball $?$.
• If I’m correct the probability of picking the $color{red}{red}$ ball at the try number $x$ is: $$
  P[X=x]=frac{(n-x)!}{n!},quad X = mbox{number of tries.}
  $$
• Thus the expected value of $X$ is:
  $$
  E(X)=sum_{x=1}^{n}xfrac{(n-x)!}{n!}
  $$
  whose ‘clean’ analytical expression I can’t find.

Could you help me advance in the problem please $?$. Unfortunately the book where I found the problem doesn’t have the solution.