\begin{align}(1+x)e^x &= e^x + xe^x = \sum^\infty_{n=0} {x^n\over n!} + \sum^\infty_{n=0} {x^{n+1}\over n!} = 1 + \sum^\infty_{n=1} {x^n\over n!} + \sum^\infty_{n=0} {x^{n+1}\over n!} \\ &= 1 + \sum^\infty_{n=1} {x^n\over n!} + \sum^\infty_{n=1} {x^{n}\over (n-1)!} =1 + \sum^\infty_{n=1}\left({1\over n!} + {1\over (n-1)!}\right)x^n \\ &= 1 + \sum^\infty_{n=1}{n+1\over n!}x^n, -\infty<x<+\infty  \\ &= \sum^\infty_{n=0}{n+1\over n!}x^n\end{align}