Contents

Definition

A power series in a variable $X$ and with coefficients in a ring $R$ is a series of the form

where $a_n$ is in $R$ for each $n\ge 0$. Given that there are no additional convergence conditions, a power series is also termed emphatically as a formal power series. If $R$ is commutative, then the collection of formal power series in a variable $X$ with coefficients in $R$ forms a commutative ring denoted by $R [ [ X ] ]$.

More generally, a power series in $k$ commuting variables $X_1,\ldots, X_k$ with coefficients in a ring $R$ has the form $\sum_{n_1=0,n_2=0,\ldots, n_k = 0}^\infty a_{n_1\ldots n_k} X_1^{n_1} X_2^{n_2}\cdots X_k^{n_k}$. If $R$ is commutative, then the collection of formal power series in $k$ commuting variables $X_1,\ldots, X_k$ form a formal power series ring denoted by $R [ [ X_1,\ldots, X_k ] ]$.

More generally, we can consider noncommutative (associative unital) ring $R$ and words in noncommutative variables $X_1,\ldots, X_k$ of the form

(where $m$ has nothing to do with $k$) and with coefficient $a_w \in R$ (here $w$ is a word of any length, not a multiindex in the previous sense). Thus the power sum is of the form $\sum_w a_w X_w$ and they form a formal power series ring in variables $X_1,\ldots, X_k$ denoted by $R\langle \langle X_1,\ldots, X_k \rangle\rangle$. Furthermore, $R$ can be even a noncommutative semiring in which case the words belong to the free monoid on the set $S = \{ X_1,\ldots, X_k\}$, the partial sums are then belong to a monoid semiring $R\langle S\rangle$. The formal power series then also form a semiring, by the multiplication rule

Of course, this implies that in a specialization, $b$-s commute with variables $X_{i_k}$; what is usually generalized to take some endomorphisms into an account (like at noncommutative polynomial level of partial sums where we get skew-polynomial rings, i.e. iterated Ore extensions).

As adic completions

Let $R$ be a commutative ring, and let $R[X]$ be the polynomial ring on one indeterminant $X$. Then $(X)$ is a maximal ideal in $R[X]$, and results in an adic topology on $R[X]$. The ring of formal power series in $R$ is the adic completion of the limit of the quotient of $R[X]$ by powers of $(X)$:

The ring of formal power series for multiple indeterminants $X_i$ is constructed iteratively: because $R[[X_1, X_2, \ldots X_n]]$ is a commutative ring, one could construct the polynomial ring $R[[X_1, X_2, \ldots X_n]][X_{n+1}]$ on the indeterminant $X_{n+1}$. As above, $(X_{n+1})$ is a maximal ideal in $R[[X_1, X_2, \ldots X_n]][X_{n+1}]$ with a corresponding adic topology, and one can then take the adic completion

The resulting commutative ring is usually just written as $R[[X_1, X_2, \ldots X_n, X_{n+1}]]$.

Examples

Polynomials

For a natural number $k$, a power series $\sum_{n=0}^\infty a_n X^n$ such that $a_n = 0$ for all $n \gt k$ is a polynomial of degree at most $k$.

Taylor series

• Taylor series

MacLaurin series

For $f \in C^\infty(\mathbb{R})$ a smooth function on the real line, and for $f^{(n)} \in C^\infty(\mathbb{R})$ denoting its $n$th derivative its MacLaurin series (its Taylor series at $0$) is the power series

If this power series converges to $f$, then we say that $f$ is analytic.

Laurent series

• Laurent series

Puiseux series

• Puiseux series

Properties

• An element $a = a_0 + a_1 x + a_2 x^2 + \ldots$ in $R[ [x] ]$ is (multiplicatively) invertible iff $a_0$ is invertible.

This follows easily from the observation that we can invert $1 + x b$ for any power series $b$ by forming $1 - x b + x^2 b^2 - \ldots$ and collecting only finitely many terms in each degree. As a simple corollary,

• If $R$ is a local ring, then the power series ring $R[ [X] ]$ is also a local ring.

Functional substitution and inversion

It follows that we may define a clone or cartesian operad as follows: the $n^{th}$ component is the set $I_n = (x_1, \ldots, x_n) \subset R[ [x_1, \ldots, x_n] ]$ which is the monad value $Ideal Pow(\{x_1, \ldots, x_n\})$. Letting $M$ denote the monad $Ideal \circ Pow$, with monad multiplication $\mu$, and $[n]$ the set $\{x_1, \ldots, x_n\}$, the clone multiplication

is the composition of the maps

The clone multiplication thus defined is called substitution of power series; it takes a tuple consisting of $p(x_1, \ldots, x_n) \in I_n, q_1(x_1, \ldots x_k) \in I_k, \ldots q_n(x_1, \ldots, x_k) \in I_k)$ to a power series denoted as

The resulting clone or operad yields, in the particular case $k = n = 1$, an associative substitution operation

with $sub(p, q) = p \circ q$ the power series $p(q(x))$.

In other words, we can functionally invert a power series provided that the linear coefficient $a_1$ is invertible in $R$.

Formal differentiation

One way to define the formal differentiation operator, as a function $\frac{\partial}{\partial X}:R[[X]] \to R[[X]]$, is via the usual formula

Then $\frac{\partial}{\partial X}$ is an $R$-linear function on $R[[X]]$ which satisfies the Leibniz rule, meaning that it is a derivation and $R[[X]]$ is a differential algebra.

Here is a conceptual story underlying the formalism. Let $D = R[\varepsilon]/(\varepsilon^2)$ be the representing object for derivations (the “ring of dual numbers”). Let $\delta: R[ [X] ] \to R[ [X] ] \otimes_R D \cong R[ [X] ][\varepsilon]/(\varepsilon^2)$ be the unique topological $R$-algebra map (under the $(X)$-adic topologies described above) that sends $X$ to $X + \varepsilon$. (If it helps, think $\delta(q) = q(X + \varepsilon)$.)

We leave as an exercise the proof that the two definitions of derivative match:

(Hint: the restriction of $p \mapsto p'$ to $R[X]$ is by construction a derivation such that $X' = 1$, and $(X^k)' = k X^{k-1}$ by induction. This induces derivations on quotient algebras $R[X]/(X^n)$, satisfying the same formula. Then pass to the inverse limit.)

See here for a conceptual proof, using the universal property of adic completion.

Relatedly but in a slightly different direction, we can consider differentiation in coalgebraic terms. Suppose the commutative ring $R$ is a commutative algebra over $\mathbb{Q}$ (thus permitting division by nonzero integers). Then the set $R[ [X]]$ may be identified with the terminal coalgebra $R^\mathbb{N}$ of the endofunctor $R \times - \colon Set \to Set$ via the map

whereby the coalgebra structure on $R^\mathbb{N}$,

corresponds to

One may then apply coinductive techniques to prove various facts. One illustration is given here, where coinduction on power series is used to prove the general binomial theorem

where, remarkably, $r$ is an arbitrary element of $R$.

Ring of power series as rings with infinitesimals

ring with infinitesimals	function
dual numbers	differentiable function
Weil ring	smooth function
power series ring	analytic function

Purely real and purely infinitesimal elements

Suppose that $K$ is a Archimedean ordered field and $K[[\epsilon]]$ is the ring of power series in $K$. Since $K[[\epsilon]]$ is a local ring, the quotient of $K[[\epsilon]]$ by its ideal of non-invertible elements $\epsilon K[[\epsilon]]$ is the residue field $K$ itself, and the canonical function used in defining the quotient is the function $\Re:K[[\epsilon]] \to K$ which takes a number $a \in K[[\epsilon]]$ to its purely real component $\Re(a) \in K$ and takes $\Re(\epsilon) = 0$. Since $K[[\epsilon]]$ is an ordered $K$-algebra, there is a strictly monotone ring homomorphism $h:K \to K[[\epsilon]]$. An element $a \in K[[\epsilon]]$ is purely real if $h(\Re(a)) = a$, and an element $a \in K[[\epsilon]]$ is purely infinitesimal if it is in the fiber of $\Re$ at $0 \in K$. Zero is the only element in $K[[\epsilon]]$ which is both purely real and purely infinitesimal.

Analytic functions

Suppose that $K$ is a sequentially Cauchy complete Archimedean ordered field with lattice structure, and $K[[\epsilon]]$ is the ring of power series of $K$. Then analytic functions are each definable on $K$ using the algebraic, order, metric, and convergence structure on $K$.

The ring homomorphism $h:K \to K[[\epsilon]]$ preserves analytic functions: given a natural number $n \in \mathbb{N}$ and a purely infinitesimal element $\eta \in \epsilon K[[\epsilon]]$, then for every analytic function $f \in C^\infty(K)$, there is a function $f_{K[[\epsilon]]}:K[[\epsilon]] \to K[[\epsilon]]$ such that for all elements $x \in K$, $f_{K[[\epsilon]]}(h(x)) = h(f(x))$ and

Related concepts

• convergence radius

• asymptotic series, transseries

• Tate algebra

• characteristic series

• power series and their formal spectra

References

• Wikipedia, Formal power series

A formalization in homotopy type theory and there in Coq is discussed in section 4 of

• Álvaro Pelayo, Vladimir Voevodsky, Michael Warren, A preliminary univalent formalization of the p-adic numbers (arXiv:1302.1207)

The discussion of the differentiation of a converging power series term by term is at

• Tim Gowers’s blog: differentiating-power-series