Multiplying and factoring in Formal Power Series

Question

I'm working with some formal power series in my homework. Somewhere in the middle of my hw problem I reach a point where I would really like to factor, but I'm not sure if I can.

Suppose $F_k$ converges to F in K[[x]]. Is $\prod_{n=1}^{\infty} F_k - x \prod_{n=1}^{\infty} F_k$ $ = (1- x) \prod_{n=1}^{\infty} F_k$? I was told to be careful when dealing with formal infinite products, but I feel like that warning was more for multiplying infinite products together.

1) Can I do this simplification? $\prod_{n=1}^{\infty} F_k - x \prod_{n=1}^{\infty} F_k = (1- x) \prod_{n=1}^{\infty} F_k$?

2) If so, how do I justify that I am allowed to do so?

See this answer for the appropriate notion of convergence. — Bill Dubuque, May 02 '16 at 20:34

score 2 · Accepted Answer · answered Apr 13 '15 at 04:09

2

The object $K[[x]]$ is often known as the ring of formal power series over a field $K$, and not for nothing — it is a ring (indeed, a commutative ring). The ring axioms imply the identity $a-ba = (1-b)a$.

answered Apr 13 '15 at 04:09

Yuval Filmus

57,157

That's what I thought. Thank you! – user3016098 Apr 13 '15 at 04:14

Multiplying and factoring in Formal Power Series

1 Answers1