# Convergence of formal power series

So I've just come across formal power series and was somewhat interested, but still can't seems to understand them. I was reading this explanation about the convergence of a formal power series and was wondering about the convergence of the following formal power series:

\begin{align} 1+x+x^2+x^3+\cdots \\ \\ 1+2x+3x^2+4x^3+\cdots \\ \\ 1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots \end{align}

From what I understand after reading this, the first formal power series is convergent because the coefficient is stable, but what about the second and third formal power series?

Is there an example where the normal power series is convergent for some value of $$x$$, but the corresponding formal power series is not convergent?

Thanks!

• The first and second series are convergent in the usual sense for some $x$ but not for other $x$; consider $x=\frac12$ and $x=2$. Are they convergent in your formal power series sense? Commented May 3, 2022 at 15:08
• Thanks, @Henry. From my (minuscule) understanding of formal power series, you're not supposed to plug in any values of $x$, and I think the first formal series is convergent, but I don't know about the second and the third though.
– mld1
Commented May 3, 2022 at 15:22
• Every series $\sum_n a_nx^n$ converges as a formal power series since the coefficients of the partial sums stabilize. If you want something that does not converge you must sum power series, such as $$\sum_k \sum_n a_{k,n}x^n.$$
– Ruy
Commented May 3, 2022 at 17:46
• The question you link to is about sequences of formal power series (dressed up as infinite sums). The notion of convergence is not relevant for formal power series themselves, Commented May 3, 2022 at 18:00
• A formal power series $$f(x)=\sum_{n=0}^{\infty} a_n x^n \tag{*}$$ always makes sense. So, there is no question of convergence here. $\text{(*)}$ is just a fancy way of rewriting the sequence $(a_0,a_1,\ldots)$ in the ring $R[[x]]$ of infinite sequences endowed with certain operations (pointwise addition and Cauchy product). However, we can endow $R[[x]]$ with the notion of convergence and then retrospectively identify $\text{(*)}$ as a true limit: $$f(x)=\lim_{N\to\infty}\sum_{n=0}^{N}a_n x^n.$$ Then again, this limit always converges since each coefficient eventually stabilizes. Commented May 3, 2022 at 19:26

A formal power series in the ring $$R$$ is a sequence $$(a_0, a_1, \ldots ),$$ where the $$a_i$$ lie in $$R$$, although this is more often denoted by $$\sum_{n=0}^\infty a_nX^n.$$ Nevertheless, all of the symbols you see above, except for the coefficients $$a_i$$ themselves, are just superfluous decorations.

The set of all formal power series is denoted by $$R[[X]]$$ and it has the structure of a ring, where you add two formal power series just by adding their respective coefficients, while the multiplication is defined by $$\Big (\sum_{n=0}^\infty a_nX^n\Big ) \Big (\sum_{n=0}^\infty b_nX^n\Big )= \Big (\sum_{n=0}^\infty c_nX^n\Big ),$$ where $$c_n:= \sum_{k=0}^n a_k b_{n-k}.$$

In addition, $$R[[X]]$$ is a topological ring by identifying it with the product topological space $$R^{\mathbb N}$$ in the obvious way, where $$R$$ is given the discrete topology.

This means that a sequence $$\Big\{\sum_{n=0}^\infty a_{n, k}X^n\Big\}_k$$ converges to an element $$\sum_{n=0}^\infty a_nX^n$$ in $$R[[X]]$$, if and only for every $$n$$, there is a $$k_0$$, such that $$a_{n,k}= a_n$$, for every $$k\geq k_0$$. In others words, if you focus on the coefficients of a single $$X^n$$, you'll notice they stop changing after a while.

Whenever you have a topological ring, you can make sense of convergence of "infinite sums", such as, $$\sum_{n=0}^\infty P_n$$, by simply interpreting it as the sequence of partial sums $$\Big\{\sum_{n=0}^N P_n\Big\}_N.$$

In the case of $$R[[X]]$$, one may choose $$P_n: = a_nX^n$$ as a formal power series in which all coefficients, except for $$a_n$$, are taken to be zero. In that case you may ask whether or not the "infinite sum" $$\sum_{n=0}^\infty a_nX^n.$$ converges in $$R[[X]]$$. Besides observing that the above is NOT to be understood as an element of $$R[[X]]$$ (it is an infinite sum of elements of $$R[[X]]$$), it is not hard to see that it does converge (Exercise: prove it!) to the element of $$R[[X]]$$ you might write as $$(a_0, a_1, \ldots )$$ in its most formal presentation.

This might be expressed by saying that every power series converges in the ring of formal power series, regardless of how big the coefficients are!

• In what sense is this a formal power series? Commented May 3, 2022 at 18:23
• Sorry @ancientmathematician, I totally rephrased my answer.
– Ruy
Commented May 3, 2022 at 19:08