power series (changes) in nLab

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Definition

A power series in a variable XX and with coefficients in a ring RR is a series of the form

∑ n=0 ∞a nX n \sum_{n = 0}^\infty a_n X^n

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

More generally, a power series in kk commuting variables X 1,…,X kX_1,\ldots, X_k with coefficients in a ring RR has the form ∑ n 1=0,n 2=0,…,n k=0 ∞a n 1…n kX 1 n 1X 2 n 2⋯X k n k\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 RR is commutative, then the collection of formal power series in kk commuting variables X 1,…,X kX_1,\ldots, X_k form a formal power series ring denoted by R[[X 1,…,X k]]R [ [ X_1,\ldots, X_k ] ].

More generally, we can consider noncommutative (associative unital) ring RR and words in noncommutative variables X 1,…,X kX_1,\ldots, X_k of the form

w=X i 1⋯X i m w = X_{i_1}\cdots X_{i_m}

(where mm has nothing to do with kk) and with coefficient a w∈Ra_w \in R (here ww is a word of any length, not a multiindex in the previous sense). Thus the power sum is of the form ∑ wa wX w\sum_w a_w X_w and they form a formal power series ring in variables X 1,…,X kX_1,\ldots, X_k denoted by R⟨⟨X 1,…,X k⟩⟩R\langle \langle X_1,\ldots, X_k \rangle\rangle. Furthermore, RR can be even a noncommutative semiring in which case the words belong to the free monoid on the set S={X 1,…,X k}S = \{ X_1,\ldots, X_k\}, the partial sums are then belong to a monoid semiring R⟨S⟩R\langle S\rangle. The formal power series then also form a semiring, by the multiplication rule

∑ ra rX r⋅∑b sX s=∑ w∑ u,v;w=uva ub vX w \sum_{r} a_r X_r \cdot \sum b_s X_s = \sum_w \sum_{u,v; w = u v} a_u b_v X_w

Of course, this implies that in a specialization, bb-s commute with variables X i kX_{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 RR be a commutative ring, and let R[X]R[X] be the polynomial ring on one indeterminant XX. Then (X)(X) is a maximal ideal in R[X]R[X], and results in an adic topology on R[X]R[X]. The ring of formal power series in RR is the adic completion of the limit of the quotient of R[X]R[X] by powers of (X)(X):

R[[X]]≔lim←R/(X) nR[[X]] \coloneqq \underset{\leftarrow}\lim R/(X)^n

The ring of formal power series for multiple indeterminants X iX_i is constructed iteratively: because R[[X 1,X 2,…X n]]R[[X_1, X_2, \ldots X_n]] is a commutative ring, one could construct the polynomial ring R[[X 1,X 2,…X n]][X n+1]R[[X_1, X_2, \ldots X_n]][X_{n+1}] on the indeterminant X n+1X_{n+1}. As above, (X n+1)(X_{n+1}) is a maximal ideal in R[[X 1,X 2,…X n]][X n+1]R[[X_1, X_2, \ldots X_n]][X_{n+1}] with a corresponding adic topology, and one can then take the adic completion

R[[X 1,X 2,…X n]][[X n+1]]≔lim←R[[X 1,X 2,…X n]]/(X n+1) mR[[X_1, X_2, \ldots X_n]][[X_{n+1}]] \coloneqq \underset{\leftarrow}\lim R[[X_1, X_2, \ldots X_n]]/(X_{n+1})^m

The resulting commutative ring is usually just written as R[[X 1,X 2,…X n,X n+1]]R[[X_1, X_2, \ldots X_n, X_{n+1}]].

Examples

Polynomials

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

Taylor series

• Taylor series

MacLaurin series

For f∈C ∞(ℝ)f \in C^\infty(\mathbb{R}) a smooth function on the real line, and for f (n)∈C ∞(ℝ)f^{(n)} \in C^\infty(\mathbb{R}) denoting its nnth derivative its MacLaurin series (its Taylor series at 00) is the power series

∑ n=0 ∞1n!f (n)(0)x n. \sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(0) x^n \,.

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

Laurent series

• Laurent series

Puiseux series

• Puiseux series

Properties

• An element a=a 0+a 1x+a 2x 2+…a = a_0 + a_1 x + a_2 x^2 + \ldots in R[[x]]R[ [x] ] is (multiplicatively) invertible iff a 0a_0 is invertible.

This follows easily from the observation that we can invert 1+xb1 + x b for any power series bb by forming 1−xb+x 2b 2−…1 - x b + x^2 b^2 - \ldots and collecting only finitely many terms in each degree. As a simple corollary,

• If RR is a local ring, then the power series ring R[[X]]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 thn^{th} component is the set I n=(x 1,…,x n)⊂R[[x 1,…,x n]]I_n = (x_1, \ldots, x_n) \subset R[ [x_1, \ldots, x_n] ] which is the monad value IdealPow({x 1,…,x n})Ideal Pow(\{x_1, \ldots, x_n\}). Letting MM denote the monad Ideal∘PowIdeal \circ Pow, with monad multiplication μ\mu, and [n][n] the set {x 1,…,x n}\{x_1, \ldots, x_n\}, the clone multiplication

I n×I k n→I kI_n \times I_k^n \to I_k

is the composition of the maps

M(n)×M(k) n≅M(n)×hom([n],M(k))→1×funcM(n)×hom(M(n),MM(k))→evalMM(k)→μ(k)M(k)M(n) \times M(k)^n \cong M(n) \times \hom([n], M(k)) \stackrel{1 \times func}{\to} M(n) \times \hom(M(n), M M(k)) \stackrel{eval}{\to} M M(k) \stackrel{\mu(k)}{\to} M(k)

The clone multiplication thus defined is called substitution of power series; it takes a tuple consisting of p(x 1,…,x n)∈I n,q 1(x 1,…x k)∈I k,…q n(x 1,…,x k)∈I k)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

p(q 1(x 1,…,x k),…q n(x 1,…,x k)).p(q_1(x_1, \ldots, x_k), \ldots q_n(x_1, \ldots, x_k)).

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

xR[[x]]×xR[[x]]→subxR[[x]]x R[ [x] ] \times x R[ [x] ] \stackrel{sub}{\to} x R[ [x] ]

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

In other words, we can functionally invert a power series provided that the linear coefficient a 1a_1 is invertible in RR.

Formal differentiation

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

∂∂X(∑ n=0 ∞a nX n)≔∑ n=0 ∞a n+1(n+1)X n. \frac{\partial}{\partial X}\left(\sum_{n = 0}^\infty a_n X^n\right) \coloneqq \sum_{n = 0}^\infty a_{n + 1} (n + 1) X^n.

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

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

\begin{definition} For p∈R[[X]]p \in R[ [X] ], the derivative p′p' is the unique element of R[[X]]R[ [X] ] satisfying

δ(p)=p(X)+p′(X)ε.\delta(p) = p(X) + p'(X)\varepsilon.

\end{definition}

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

p′(X)=∂∂Xp(X).p'(X) = \frac{\partial}{\partial X} p(X).

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

\begin{proposition} (Chain rule) For p∈R[[X]]p \in R[ [X] ] and q∈xR[[X]]q \in x R[ [X ] ],

(p∘q)′=(p′∘q)⋅q′.(p \circ q)' = (p' \circ q) \cdot q'.

\end{proposition}

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 RR is a commutative algebra over ℚ\mathbb{Q} (thus permitting division by nonzero integers). Then the set R[[X]]R[ [X]] may be identified with the terminal coalgebra R ℕR^\mathbb{N} of the endofunctor R×−:Set→SetR \times - \colon Set \to Set via the map

R[[X]]→R ℕ:∑ n≥0a nX nn!↦(a 0,a 1,a 2,…)R[ [X]] \to R^\mathbb{N}\; : \; \sum_{n \geq 0} \frac{a_n X^n}{n!} \mapsto (a_0, a_1, a_2, \ldots)

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

R ℕ→R×R ℕ:(a n) n≥0↦⟨a 0,(a n+1) n≥0⟩,R^\mathbb{N} \to R \times R^\mathbb{N}\; \colon \; (a_n)_{n \geq 0} \mapsto \langle a_0, (a_{n+1})_{n \geq 0} \rangle,

corresponds to

R[[X]]→R×R[[X]]:f(X)↦⟨f(0),f′(X)⟩.R[ [X]] \to R \times R[ [X]]\; \colon\; f(X) \mapsto \langle f(0), f'(X) \rangle.

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

(1+x) r≔exp(rlog(1+x))=∑ k≥0r k̲x kk!(1 + x)^r \coloneqq \exp(r \log(1 + x)) = \sum_{k \geq 0} \frac{r^\underline{k} x^k}{k!}

where, remarkably, rr is an arbitrary element of RR.

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 KK is a Archimedean ordered field and K[[ϵ]]K[[\epsilon]] is the ring of power series in KK. Since K[[ϵ]]K[[\epsilon]] is a local ring, the quotient of K[[ϵ]]K[[\epsilon]] by its ideal of non-invertible elements ϵK[[ϵ]]\epsilon K[[\epsilon]] is the residue field KK itself, and the canonical function used in defining the quotient is the function ℜ:K[[ϵ]]→K\Re:K[[\epsilon]] \to K which takes a number a∈K[[ϵ]]a \in K[[\epsilon]] to its purely real component ℜ(a)∈K\Re(a) \in K and takes ℜ(ϵ)=0\Re(\epsilon) = 0. Since K[[ϵ]]K[[\epsilon]] is an ordered KK-algebra, there is a strictly monotone ring homomorphism h:K→K[[ϵ]]h:K \to K[[\epsilon]]. An element a∈K[[ϵ]]a \in K[[\epsilon]] is purely real if h(ℜ(a))=ah(\Re(a)) = a, and an element a∈K[[ϵ]]a \in K[[\epsilon]] is purely infinitesimal if it is in the fiber of ℜ\Re at 0∈K0 \in K. Zero is the only element in K[[ϵ]]K[[\epsilon]] which is both purely real and purely infinitesimal.

Analytic functions

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

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

f K[[ϵ]](h(x)+η)=∑ i=0 ∞1i!h(d ifdx i(x))η if_{K[[\epsilon]]}(h(x) + \eta) = \sum_{i = 0}^{\infty} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}(x)\right) \eta^i

• 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