The sum of the infinite geometric series $1+x+x^2+x^3+\dots$ is well known.
Show that

${\displaystyle \sum _{n=0}^{\infty }n{x}^n=\frac{x}{(1-x{)}^2}}$

and find

${\displaystyle \sum _{n=0}^{\infty }{n}^2{x}^n.}$

Outline a method for finding

${\displaystyle \sum _{n=0}^{\infty }{n}^k{x}^n}$

where you do not have to carry out this computation beyond $k=2$.