Given any $\mathbb{Z}_{(p)}$-algebra $R$, we can form the big Witt vectors $(a_n) \in \prod_{n=1}^\infty R$. Then there is an isomorphism from $$BigWitt(R) \to 1+xR[[x]].$$ I believe one can define the addition and multiplication of the big witt vectors as the coordinate wise addition and multiplication of their ghost components. However, all references I can find are only for the standard Witt vectors.
So my question is, is this true and, if so, is there a simple way to write down the ghost components of the big Witt vectors?