# function representation of power series

What is the function representation of this power series?

[Summation from n=0 to infinity of ($x^n)(n+1)!/n!$

The solution is $\frac{1}{(1-x)^-2}$ but how???

I know that $\sum_{n=0}^{\infty}(x^n)/n! = e^x$, but I don't know how to get to the solution from there.

Notice that $$\frac{(n+1)!}{n!}x^n=\frac{(n+1)n!}{n!}x^n=(n+1)x^n$$ So, $$\sum_{n=0}^\infty\frac{(n+1)!}{n!}x^n=\sum_{n=0}^\infty (n+1)x^n=\sum_{n=0}^\infty nx^n +\sum_{n=0}^\infty x^n=x\sum_{n=0}^\infty nx^{n-1} +\sum_{n=0}^\infty x^n=$$ $$x \frac d {dx}\Big( \sum_{n=0}^\infty x^n\Big)+\sum_{n=0}^\infty x^n$$
• Remember that $k!=k\times (k-1)!=k\times (k-1)\times (k-2)!$ – Claude Leibovici Apr 6 '16 at 6:13
• What is $\sum_{n=0}^\infty x^n$ ? – Claude Leibovici Apr 6 '16 at 6:23