[proofplan]
We treat the infinite sum as a formal [power series](/page/Power%20Series), not as an analytically convergent series. Define $S(X)$ coefficientwise, multiply it by $1_R-X$ using the formal Cauchy product rule in $R[[X]]$, and compute each coefficient of the product. The constant coefficient is $1_R$, while every positive-degree coefficient cancels as $1_R-1_R=0_R$; coefficientwise equality then identifies the product with the constant series $1_R$.
[/proofplan]
[step:Compute the coefficients of the product $(1_R-X)S(X)$]
Define $A(X)\in R[[X]]$ by
\begin{align*}
A(X)=1_R-X.
\end{align*}
Thus
\begin{align*}
[X^0]A(X)=1_R,
\end{align*}
\begin{align*}
[X^1]A(X)=-1_R,
\end{align*}
and
\begin{align*}
[X^i]A(X)=0_R
\end{align*}
for every integer $i\geq 2$. Define $C(X)\in R[[X]]$ by
\begin{align*}
C(X)=A(X)S(X).
\end{align*}
By the definition of multiplication in the formal power series ring, for every integer $m\geq 0$,
\begin{align*}
[X^m]C(X)=\sum_{i=0}^{m}[X^i]A(X)[X^{m-i}]S(X).
\end{align*}
For $m=0$, this gives
\begin{align*}
[X^0]C(X)=[X^0]A(X)[X^0]S(X)=1_R.
\end{align*}
For $m\geq 1$, the only possibly nonzero summands are the terms with $i=0$ and $i=1$, so
\begin{align*}
[X^m]C(X)=[X^0]A(X)[X^m]S(X)+[X^1]A(X)[X^{m-1}]S(X).
\end{align*}
Since $[X^k]S(X)=1_R$ for every integer $k\geq 0$, we obtain
\begin{align*}
[X^m]C(X)=1_R\cdot 1_R+(-1_R)\cdot 1_R=0_R.
\end{align*}
[guided]
The point is that formal power series multiplication is coefficientwise finite: the coefficient of $X^m$ in a product only depends on finitely many pairs of coefficients whose degrees add to $m$. Define the auxiliary series $A(X)\in R[[X]]$ by
\begin{align*}
A(X)=1_R-X.
\end{align*}
Its coefficients are
\begin{align*}
[X^0]A(X)=1_R,
\end{align*}
\begin{align*}
[X^1]A(X)=-1_R,
\end{align*}
and
\begin{align*}
[X^i]A(X)=0_R
\end{align*}
for every integer $i\geq 2$. Also, by definition of $S(X)$,
\begin{align*}
[X^k]S(X)=1_R
\end{align*}
for every integer $k\geq 0$.
Now define the product series $C(X)\in R[[X]]$ by
\begin{align*}
C(X)=A(X)S(X).
\end{align*}
By the formal Cauchy product rule, for each integer $m\geq 0$ the coefficient of $X^m$ in $C(X)$ is
\begin{align*}
[X^m]C(X)=\sum_{i=0}^{m}[X^i]A(X)[X^{m-i}]S(X).
\end{align*}
This sum is finite, so no convergence issue is involved.
For the constant coefficient, $m=0$, there is only one summand:
\begin{align*}
[X^0]C(X)=[X^0]A(X)[X^0]S(X)=1_R\cdot 1_R=1_R.
\end{align*}
For a positive degree $m\geq 1$, all summands with $i\geq 2$ vanish because $[X^i]A(X)=0_R$. Therefore only the terms $i=0$ and $i=1$ remain:
\begin{align*}
[X^m]C(X)=[X^0]A(X)[X^m]S(X)+[X^1]A(X)[X^{m-1}]S(X).
\end{align*}
Substituting the coefficients of $A(X)$ and $S(X)$ gives
\begin{align*}
[X^m]C(X)=1_R\cdot 1_R+(-1_R)\cdot 1_R.
\end{align*}
Because $-1_R$ is the additive inverse of $1_R$ in $R$, this coefficient is
\begin{align*}
[X^m]C(X)=0_R.
\end{align*}
Thus the product has constant coefficient $1_R$ and every positive-degree coefficient equal to $0_R$.
[/guided]
[/step]
[step:Identify the product with the constant series $1_R$]
Let $E(X)\in R[[X]]$ denote the constant formal power series $1_R$. Then
\begin{align*}
[X^0]E(X)=1_R
\end{align*}
and
\begin{align*}
[X^m]E(X)=0_R
\end{align*}
for every integer $m\geq 1$. The coefficient computation above gives
\begin{align*}
[X^m]C(X)=[X^m]E(X)
\end{align*}
for every integer $m\geq 0$. Since $C(X),E(X)\in R[[X]]$, equality in $R[[X]]$ is coefficientwise equality of sequences indexed by the integers $m\geq 0$. The displayed coefficient identity therefore implies
\begin{align*}
C(X)=E(X).
\end{align*}
Substituting back the definitions of $C(X)$, $A(X)$, and $E(X)$, we obtain
\begin{align*}
(1_R-X)S(X)=1_R.
\end{align*}
Since $S(X)=\sum_{n=0}^{\infty}X^n$, this is exactly
\begin{align*}
(1_R-X)\sum_{n=0}^{\infty}X^n=1_R.
\end{align*}
[guided]
We now translate the coefficient computation into equality of formal power series. Let $E(X)\in R[[X]]$ be the constant formal power series $1_R$. By definition,
\begin{align*}
[X^0]E(X)=1_R
\end{align*}
and
\begin{align*}
[X^m]E(X)=0_R
\end{align*}
for every integer $m\geq 1$. The preceding coefficient computation showed exactly that $C(X)$ has the same coefficients: the constant coefficient is $1_R$, and every positive-degree coefficient is $0_R$. Hence
\begin{align*}
[X^m]C(X)=[X^m]E(X)
\end{align*}
for every integer $m\geq 0$.
A formal power series in $R[[X]]$ is determined by its coefficient sequence indexed by the integers $m\geq 0$. Since both $C(X)$ and $E(X)$ are elements of $R[[X]]$ and their coefficients agree in every degree, we get
\begin{align*}
C(X)=E(X).
\end{align*}
Finally, $C(X)=A(X)S(X)$, $A(X)=1_R-X$, and $E(X)$ is the constant formal power series $1_R$. Substituting these definitions gives
\begin{align*}
(1_R-X)S(X)=1_R.
\end{align*}
Using the defining notation $S(X)=\sum_{n=0}^{\infty}X^n$, this is the claimed identity
\begin{align*}
(1_R-X)\sum_{n=0}^{\infty}X^n=1_R.
\end{align*}
[/guided]
[/step]
