# nLab power series

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

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

$\sum _{n=0}^{\infty }{a}_{n}{X}^{n}$\sum_{n = 0}^\infty a_n X^n

with coefficients $\left({a}_{n}\in R{\right)}_{n=0}^{\infty }$. If there are no additional convergence conditions on a power series we call it for emphasis also formal power series.

If there is $k\in ℕ$ such that ${a}_{n}=0$ for all $n>k$ then this is a polynomial of degree $k$.

The collection of formal power series in variable $X$ with coefficients in a commutative ring $R$ is denoted $R\left[\left[X\right]\right]$.

More generally, one considers power series ${\sum }_{{n}_{1}=0,{n}_{2}=0,\dots ,{n}_{k}=0}^{\infty }{a}_{{n}_{1}\dots {n}_{k}}{X}_{1}^{{n}_{1}}{X}_{2}^{{n}_{2}}\cdots {X}_{k}^{{n}_{k}}$ in $k$ variables ${X}_{1},\dots ,{X}_{k}$ which are declared commutative with ${a}_{{n}_{1}\dots {n}_{k}}\in R$, where $R$ is commutative; they form a formal power series ring $R\left[\left[{X}_{1},\dots ,{X}_{k}\right]\right]$. More generally, we can consider noncommutative (associative unital) ring $R$ and words in noncommutative variables ${X}_{1},\dots ,{X}_{k}$ of the form

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

(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 ${X}_{1},\dots ,{X}_{k}$ denoted by $R⟨⟨{X}_{1},\dots ,{X}_{k}⟩⟩$. Furthermore, $R$ can be even a noncommutative semiring in which case the words belong to the free monoid on the set $S=\left\{{X}_{1},\dots ,{X}_{k}\right\}$, the partial sums are then belong to a monoid semiring $R⟨S⟩$. The formal power series then also form a semiring, by the multiplication rule

$\sum _{r}{a}_{r}{X}_{r}\cdot \sum {b}_{s}{X}_{s}=\sum _{w}\sum _{u,v;w=uv}{a}_{u}{b}_{v}{X}_{w}$\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, $b$-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).

## Examples

### MacLaurin series

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

$\sum _{n=0}^{\infty }\frac{1}{n!}{f}^{\left(n\right)}\left(0\right){x}^{n}\phantom{\rule{thinmathspace}{0ex}}.$\sum_{n = 0}^\infty \frac{1}{n!} f^{(n)}(0) x^n \,.

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

## References

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

Revised on November 5, 2013 23:25:32 by Urs Schreiber (82.169.114.243)