Sum of infinite power series with factorial

Question

I don't know how to start to solve this, can you give me some hints? I can solve it without the factorial, but with the factorial, it creates problems...

Thank you!

Answer 1

Write down the Maclaurin series for $xe^x$ and differentiate it wrt $x$.

Answer 2

$$\dfrac n{(n-1)!}x^n=x^2\cdot\dfrac{x^{n-2}}{(n-2)!}+x\cdot\dfrac{x^{n-1}}{(n-1)!}$$

$e^y=\sum_{r=0}^\infty\dfrac{y^r}{r!}$