I am currently attempting to read Hazelwinkel's paper "Witt Vectors 1", in which the ring of big Witt vectors $W(R)$ for $R$ a commutative unital ring is defined to be the set $R^\mathbb{N}$ with ring structure induced by the bijection:
$$ e_R: (x_1, x_2, \dots ) \mapsto \prod_{i=1}^\infty (1 - x_it^i)^{-1} $$
between $W(R)$ and the set $\Lambda(R) := 1 + tR[[t]]$ of all power series in $R$ with constant term $1$, where addition is power series multiplication, and multiplication is another operation defined in the paper.
My question simply is: why is the above actually a bijection? I have tried working this out, and it is clear that the above is injective, but I do not see why it should in general be surjective.
I am not entirely sure what tags should be used, here.
On the right hand side the $t^n$ term is the sum of all $x_1^{i_1}\cdots x_n^{i_n}t^n$ such that $\sum k\cdot i_k=n$. For $n=1$ this is $x_1$, for $n=2$ this is $x_1^2 + x_2$, in general it is $x_n + f(x_1,\ldots,x_{n-1})$. So to get a power series $\sum a_it^i$ set $x_1 = a_1$ and $x_2 = a_2 -x_1^2$ and in general $x_n = a_n - f(x_{<n})$.