I recently discovered the following cute formal noncommutative power series identity: if $(x_i)_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series $$ 1 + \sum_{m=1}^\infty \sum_i x_i^m = 1 + \sum_i x_i+ \sum_i x_i^2 + \sum_i x_i^3 + \dots$$ is the reciprocal of the formal power series $$ \sum_{k=0}^\infty (-1)^k \sum_{i_1 \neq \dots \neq i_k} x_{i_1} \dots x_{i_k}$$ $$ = 1 - \sum_i x_i + \sum_{i \neq j} x_i x_j - \sum_{i \neq j \neq k} x_i x_j x_k + \dots$$ where summation indices are understood to range in $I$ if not otherwise specified. (Note that we do not require the $i_1,\dots,i_k$ to all be distinct from each other; it is only consecutive indices $i_j, i_{j+1}$ that are required to be distinct. So this isn't just the Newton identities relating power sums with elementary symmetric polynomials, though it seems to be a cousin of these identities.)

For instance, if $|I|=n$ and $x_i=x$, this identity amounts (after summing the geometric series) to the (formal) assertion $$ (1 + \frac{nx}{1-x})^{-1} = 1 - \frac{nx}{1+(n-1)x}$$ which follows from high school algebra.

Once written down, the general identity is not hard to prove: multiply the two power series together and observe that every non-constant term with a coefficient of $+1$ is cancelled by a term with a coefficient of $-1$ and vice versa. But I am certain that an identity this basic must already be in either the enumerative combinatorics or the physics literature. Does it have a name, and where is it used? Presumably there is also some natural categorification (or at least a bijective or probabilistic proof).