How to represent the given series in exponential form

Question

I was working on a problem that devolved to the summation $\Sigma_{n=1}^\infty n \dfrac{x^{n-1}}{(n-1)!}$. Is there any exponential representation available for such a term?

Answer 1

Write the top $n$ as $n = (n-1)+1$ to get that the terms add up to $(1+x) e^x$.

Answer 2

$$ \int (\Sigma_{n=1}^\infty n \dfrac{x^{n-1}}{(n-1)!}) dx = \Sigma_{n=1}^\infty \int n \dfrac{x^{n-1}}{(n-1)!} dx = x \sum_{n=0} x^n/n! = xe^x $$ Now differentiate to get result.