Closed Form for $\sum{\frac{x^{n-1}}{n!}}$

Question

I need a nicer form of $$\sum_{n=0}^{\infty}\frac{x^{n-1}}{n!}$$ I know the typical trick of recognizing the inside as the derivative or integral of something and then moving the sum on the inside of the expression but it doesn't work here. It can be seen as the generating function of $$a_n = \frac{1}{n+1}$$ Thanks.

Answer 1

$$\sum_{n=0}^{\infty}\frac{x^{n-1}}{n!}=\frac1x\sum_{n=0}^{\infty}\frac{x^{n}}{n!}$$ Can you proceed from there?