For the following function, find the full power series centered at $x=0$ and then give the first $5$ nonzero terms of the power series and the open interval of convergence.

$f(x)=\frac{x^4}{(1+4x{)}^2}$

$f(x)={\displaystyle \underset{n=0}{\overset{}{}}}$

$f(x)=+++++++$cdots

The open interval of convergence is: $($Give your answer in help $($intervals$).).$