For this question, I am asked to express $\sum\_{n=1}^{\infty}\frac{nx^n}{(n+1)!}$ as elementary functions (for example, $\log x, x^2, \sqrt{x}$, etc)

List of elementary functions v

<https://en.wikipedia.org/wiki/Elementary_function#Examples>

How would I break down an infinite sum into such functions? Well, first thing I thought about is using integration techniques. We can write the infinite sum as$\sum\_{n=1}^{\infty}\frac{n}{(n+1)!}x^n$. Let $a\_n = \frac{n}{(n+1)!}$, then $\sum\_{n=0}^{\infty}\int a\_n x^ndx = \sum\_{n=1}^{\infty}\frac{a\_n}{n+1} x^{n+1}+C$, and C is a constant. Is this approach correct? If so, how would it work?