There is an identity $e^x=\lim_{n\to \infty} (1+x/n)^n$, and I always thought it is a purely analytic statement. But then I discovered its curious interpretation in pure algebra:

Consider the ring of formal infinite sums of monomials in infinitely many variables $\varepsilon_1, \varepsilon_2,\ldots$ satisfying $\varepsilon_i^2=0$.

$$ R=\mathbb{Q}[\![\varepsilon_1, \varepsilon_2, \ldots]\!]/(\varepsilon_i^2: i=1,2,\ldots). $$

Then the sum $x=\sum_{i=1}^\infty \varepsilon_i$ makes sense and is not infinitesimally small, in fact we have $$ x^n = n! \sum_{1\leq i_1<i_2<\ldots<i_n} \varepsilon_{i_1} \cdots \varepsilon_{i_n}. $$ So the ring of polynomials $\mathbb{Q}[x]$ embeds into $R$. Moreover, in $R$ we have the identity $$ \prod_{i=1}^\infty(1+\varepsilon_i) = \sum_{n=0}^\infty \frac{x^n}{n!}. $$ So somehow we multiplied infinitely many elements infinitely close to $1$ and managed to get away from $1$ and obtain the right answer.

I was wondering if this is well-known and if there are applications of this idea. For instance, one can probably use it to recover the formal neighborhood of $1$ in an algebraic group from the Lie algebra.

In positive characteristic the right hand side doesn't make sense, but the left hand side still does. In fact, symmetric functions in $\{\varepsilon_i\}$ form a ring with divided power structure. Can one build $p$-adic cohomology theories based on this idea instead of divided power structures?