Let me work out the case of an infinite index set $I$. Let $R$ be a ring with unit. Then $$U := R[[x_i| i \in I]]$$ can be seen as the union of the "finite" $R[[x_{i_1}, \dots, x_{i_n}]]$ with $i_1, \dots, i_n \in I$.

Now suppose $f\in U$ is invertible. So there is $g \in U$ such that $fg=1$. Moreover we can find a large enough $n$ such that $f,g \in R[[x_{i_1}, \dots, x_{i_n}]]$. From the finite case it follows that

$f \in R^{\times} + (x_{i_1}, \dots, x_{i_n}) \le R^{\times} + (x_i | i \in I) =: P$.

Conversely suppose $f \in P$. Then $f \in R^{\times} + (x_{i_1}, \dots, x_{i_n})$ for some $n$. Thus there is $g \in R[[x_{i_1}, \dots, x_{i_n}]] \le U$ such that $fg=1$. So $f$ is invertible in $U$, showing $U^{\times} = P$.

Most properties of $U$ can be derived in this way.

Edit: As pointed out by unknown (google) and Andreas Blass it's not true that the power series ring is in general a direct limit of it's subrings in finitely many variables. So what I have written below only holds in the subring $\varinjlim_J R[[x_j|j \in J]]$ where $J$ runs through the finite subsets of $I$.

Let me work out the case of an infinite index set $I$. Let $R$ be a ring with unit. Then $$U := R[[x_i| i \in I]]$$ can be seen as the union of the "finite" $R[[x_{i_1}, \dots, x_{i_n}]]$ with $i_1, \dots, i_n \in I$.

Now suppose $f\in U$ is invertible. So there is $g \in U$ such that $fg=1$. Moreover we can find a large enough $n$ such that $f,g \in R[[x_{i_1}, \dots, x_{i_n}]]$. From the finite case it follows that

$f \in R^{\times} + (x_{i_1}, \dots, x_{i_n}) \le R^{\times} + (x_i | i \in I) =: P$.

Conversely suppose $f \in P$. Then $f \in R^{\times} + (x_{i_1}, \dots, x_{i_n})$ for some $n$. Thus there is $g \in R[[x_{i_1}, \dots, x_{i_n}]] \le U$ such that $fg=1$. So $f$ is invertible in $U$, showing $U^{\times} = P$.

Most properties of $U$ can be derived in this way.