Can a formal sum be 'infinite'?

Question

I'm reading Abstract Algebra (by Doomit and Foote), where a polynomial is defined as the formal sum

$$a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$$

with $n \geq 0$.

Is it valid to define a polynomial as $\sum_{j=0}^{\infty} a_jx^j$, for which exists an $n$ such that $m \geq n$ implies $a_m=0$, and said that it is a formal sum?

Bernard · Accepted Answer · 2021-02-11T22:06:09.630

3

Yes, it can.

The abstract definition of a polynomial with coefficients in a commutative ring $R$ is ‘an element of the set $R^{(\mathbf N)}$, i.e. an infinite sequence of elements of $R$ with finite support. I has a structure of an $R$-algebra if addition is defined componentwise and multiplication is defined as a Cauchy product. With this approach, $X$ denotes the sequence $(0,1,0,0,\dots,0,\dots)$, and one easily checks that $X^2=(0,0,1,0,\dots,0,\dots)$, $X^3=(0,0,0,1,0,\dots)$, and so on.

Similarly, one defines the ring of formal power series, denoted $R[[X]]$, as the set $R^{\mathbf N}$, i.e. the set of infinite sequences without any condition, endowed with addition componentwise and the Cauchy product. The ring of polynomials $R[X]$ is a subalgebra of the $R$-algebra $R[[X]]$.

edited Feb 11 '21 at 22:06

answered Feb 11 '21 at 21:52

Bernard

175,478

can it be called a 'formal sum' too? – John Mars Feb 11 '21 at 21:59
@JohnMars It's usually called a formal power series, but formal sum will do. – Jean-Claude Arbaut Feb 11 '21 at 22:06
It can, but the rigourous definition uses what I wrote (e.g. in Bourbaki). If the sequence with finite support has $0$ terms for $n>d$, the polynomial is usually written as $a_0+a_1X+\dots+a_d X^d$. For a formal power series, one usually writes $\sum_{i\ge0}a_iX^i$. – Bernard Feb 11 '21 at 22:09

Can a formal sum be 'infinite'?

1 Answers1