Completion of perfection of Laurent series

Question

Let $k/\mathbb{F}_p$ be perfect. I have read that $K:=k((t^{1/p^\infty}))$ is defined as the completion of the perfection of $k((t))$. Usually $k((t^{1/p^\infty}))$ is defined as $k((t))[t^{1/p}, t^{1/p^2}, \dots]$, but this is just the perfection (perfect closure) of $k((t))$. My question is: Is $k((t))[t^{1/p}, t^{1/p^2}, \dots]$ already complete? And if not so, how exactly does the completion $K$ look like?