\sum_{n=0}^ \infty n \frac{x^{n}}{n!}= \sum_{n=1}^ \infty n \frac{x^{n}}{n!}=\sum_{n=1}^ \infty \frac{x^{n}}{(n-1)!}=x \sum_{n=1}^ \infty \frac{x^{n-1}}{(n-1)!}=x \sum_{n=0}^ \infty \frac{x^{n}}{n!}=xe^{x}