There are related examples at this MO question, but most power series identities can be categorified to natural isomorphisms between combinatorial species, which are functors $\text{FinSet}_0 \to \text{FinSet}_0$ from the category of finite sets and bijections to itself. The idea is that the decategorification of a species $F$ is the power series $\sum F(n) \frac{x^n}{n!}$ where $F(n)$ is the cardinality of $F(S)$, where $|S| = n$.

Then $x$ is the decategorification of the species $X$ which corresponds to the structure of "being a one-element set." $L = \frac{1}{1-x}$ is the decategorification of the unique species satisfying $L \cong 1 + xL$, or "an $L$-structure is either empty or an $x$-structure together with an $L$-structure." (Addition and multiplication of generating functions correspond to natural operations on species which are left as an exercise to define.) Then the identity we want is $L \cong 1 + x + x^2 + ...$ which follows just by repeatedly substituting the isomorphism $L \cong 1 + xL$ into itself.

Alternately one can define $L$ to be the species of linear orders and then show that $L \cong 1 + xL$.

The finite case is similar. Of course one can go much further with these ideas; see, for example, Bergeron, Labelle, and Leroux. I am sure Todd Trimble will also have something interesting to say.

There are related examples at this MO question, but most power series identities can be categorified to natural isomorphisms between combinatorial species, which are functors $\text{FinSet}_0 \to \text{FinSet}_0$ from the category of finite sets and bijections to itself. The idea is that the decategorification of a species $F$ is the power series $\sum F(n) \frac{x^n}{n!}$ where $F(n)$ is the cardinality of $F(S)$, where $|S| = n$.

Then $x$ is the decategorification of the species $X$ which corresponds to the structure of "being a one-element set." $L = \frac{1}{1-x}$ is the decategorification of the unique species satisfying $L \cong 1 + xL$, or "an $L$-structure is either empty or an $x$-structure together with an $L$-structure." (Addition and multiplication of generating functions correspond to natural operations on species which are left as an exercise to define.) Then the identity we want is $L \cong 1 + x + x^2 + ...$ which follows just by repeatedly substituting the isomorphism $L \cong 1 + xL$ into itself.

Alternately one can define $L$ to be the species of linear orders and then show that $L \cong 1 + xL$.

The finite case is similar. Of course one can go much further with these ideas; see, for example, Bergeron, Labelle, and Leroux. I am sure Todd Trimble will also have something interesting to say.

There are related examples at this MO question, but most power series identities can be categorified to natural isomorphisms between combinatorial species, which are functors $\text{FinSet}_0 \to \text{FinSet}_0$ from the category of finite sets and bijections to itself. The idea is that the decategorification of a species $F$ is the power series $\sum F(n) \frac{x^n}{n!}$ where $F(n)$ is the cardinality of $F(S)$, where $|S| = n$.

Then $x$ is the decategorification of the species $X$ which corresponds to the structure of "being a one-element set." $L = \frac{1}{1-x}$ is the decategorification of the unique species satisfying $L \cong 1 + xL$, or "an $L$-structure is either empty or an $x$-structure together with an $L$-structure." (Addition and multiplication of generating functions correspond to natural operations on species which are left as an exercise to define.) Then the identity we want is $L \cong 1 + x + x^2 + ...$ which follows just by repeatedly substituting the isomorphism $L \cong 1 + xL$ into itself.

The finite case is similar. Of course one can go much further with these ideas; see, for example, Bergeron, Labelle, and Leroux. I am sure Todd Trimble will also have something interesting to say.

There are related examples at this MO question, but most power series identities can be categorified to natural isomorphisms between combinatorial species, which are functors $\text{FinSet}_0 \to \text{FinSet}_0$ from the category of finite sets and bijections to itself. The idea is that the decategorification of a species $F$ is the power series $\sum F(n) \frac{x^n}{n!}$ where $F(n)$ is the cardinality of $F(S)$, where $|S| = n$.

Then $x$ is the decategorification of the species $X$ which corresponds to the structure of "being a one-element set." $L = \frac{1}{1-x}$ is the decategorification of the unique species satisfying $L \cong 1 + xL$, or "an $L$-structure is either empty or an $x$-structure together with an $L$-structure." (Addition and multiplication of generating functions correspond to natural operations on species which are left as an exercise to define.) Then the identity we want is $L \cong 1 + x + x^2 + ...$ which follows just by repeatedly substituting the isomorphism $L \cong 1 + xL$ into itself.

Alternately one can define $L$ to be the species of linear orders and then show that $L \cong 1 + xL$.

The finite case is similar. Of course one can go much further with these ideas; see, for example, Bergeron, Labelle, and Leroux. I am sure Todd Trimble will also have something interesting to say.