Let $K$ be a perfect field of characteristic $p$, with $p>0$, which is complete with respect to a non-archimedean absolute value (in particular, elements of $\mathbb{F}_p$ have absolute value 0 or 1). Let $W$ be the associated ring of Witt vectors. We know $W$ is complete with respect to the $p$-adic topology and that every element of $W$ can be written uniquely as $$x=\sum_{i=0}^{\infty}\chi(x_i)p^i$$ where $\chi:K\to W$ is the Teichmuller character and $x_i\in k$ for every $i$. This also allows to see $W$ as the ring $K^{\mathbb{N}}$ via the identification $x\to (x_0,x_1^p,x_2^{p^2},\ldots,)$, but where addition and multiplication are not componentwise, instead they respect the rules given by Witt polynomials. From now on, I will always work in this second setting.
My question is the following: consider a series of elements of $W$ of the form $$\sum_{j=0}^{\infty}w_j = \sum_{j=0}^{\infty}(w_{i,j})_{i\in\mathbb{N}}.$$ Is it true that the convergence in $K$ of the series $\sum_{j=0}^{\infty}w_{i,j}$ for every $i$ yields convergence of the series $\sum_{j=0}^{\infty}w_j$ in $W$?
It is immediate to see that the 0-component of the series is equal to $\sum_{j=0}^{\infty}w_{0,j}$. The 1-component is sufficiently simple to guarantee that it is given by $\sum_{j=0}^{\infty}w_{1,j}$ minus a series with terms of the form $b\cdot(w_{0,0}+\cdots+w_{0,n})^{a/p}\cdot w_{0,n+1}^{(p-a)/p}$, where $1\leq a\leq p-1$ and $b\in\mathbb{F}_p$. I expect that a similar behavior occurs in every other component, but the Witt relations seem to become too complicated to give an explicit expression like the previous ones.
Concluding, does anyone know if this claim is true? Is there any precise reference for this?
Thank you in advance.