Find the expression for the sum of this power series: $\sum_{n=0}^{\infty} \frac{x^{n}}{(n+3)!} $

Question

I'm a bit stuck on this problem. I supposed to find an expression for the sum of this power series

$$\sum_{n=0}^{\infty} \frac{x^{n}}{(n+3)!} $$

I know I should use $$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!} $$

But then I get confused, and I don't how I shall proceed to find the answer.

Answer 1

$$\sum_{n=0}^{\infty} \frac{x^{n}}{(n+3)!}=\sum_{n=3}^{\infty} \frac{x^{n-3}}{n!}=x^{-3}\sum_{n=3}^{\infty} \frac{x^{n}}{n!}=x^{-3}(e^x-\frac{x^0}{0!}-\frac{x^1}{1!}-\frac{x^2}{2!})=\frac{e^x}{x^3}-\frac{1}{x^3}-\frac{1}{x^2}-\frac{1}{2x}$$ I believe that I've done everything correctly.