Unclear definition concerning convergence of a power series

Question

I am following a course on computer algebra and at the end of the course, my professor wrote something down I could not follow at first. Concretely it handles about a definition given as follows:

Let $(b_n(x))_{n \geq 0}$ be a sequence of power series $\in \mathbb{K}[[x]]$.
Set $a_N := \sum_{n=0}^N b_n(x) \in \mathbb{K}[[x]]$ with $N \in \mathbb{N}$.

If $(a_N)_{N \geq 0}$ is convergent, i.e. $\exists b(x) \in \mathbb{K}[[x]]$ such that $\lim_\limits{N \to \infty} a_N(x) = b(x) = \lim_\limits{N \to \infty} \sum_{n=0}^N b_n(x) = \lim_\limits{n \to \infty} a_N(x) = b(x)$,
then we define $\sum_{n=0}^\infty b_n := b(x)$ $(= \lim_\limits{N \to \infty} \sum_{n = 0}^N b_n)$

Note that $\mathbb{K}[[x]]$ represents the ring of power series with Cauchy product and $\lim_\limits{n \to \infty} a_n(x)$ was introduced as notation for a convergent power series $a_n(x)$.

I tried to understand what the goal of this definition should be, but it is written down in such a confusing and sloppy way that I do not seem to get the sense of it anymore. I know we are working towards composition of formal power series, but I don't see why.

Because I need to solve some homeworks from this definition, I was wondering whether any one could guide me to references concerning this definition (e.g. does it have a name?) or explain what the goal of this definition could be.

Answer 1

The definition may become easier to follow if we instead consider a comparable, more familiar situation.

Let's consider a real sequence $(b_n)_{n\geq 0}$ and the series \begin{align*} \sum_{n=0}^\infty b_n\tag{1} \end{align*} The series is defined in two ways. At one hand it is defined as sequence, namely the sequence of it's partial sums \begin{align*} \sum_{n=0}^\infty b_n:=\left(\sum_{n=0}^Nb_n\right)_{N\geq 0}\tag{2} \end{align*} on the other hand it is defined as value. If the limit of the sequence of partial sums converges to a real value, we say the series (1) converges and define it's value as \begin{align*} \sum_{n=0}^{\infty} b_n:= \lim_{N\rightarrow \infty}\left(\sum_{n=0}^Nb_n\right)\tag{3} \end{align*}

You could find here a somewhat more detailed answer regarding this aspect.

In the framework of formal power series we can find similar definitions. We consider instead of real sequences a sequence of elements from $\mathbb{K}[[x]]$ which is $(b_n(x))_{n\geq 0}$ and want to define the series \begin{align*} \sum_{n=0}^\infty b_n(x)\tag{4} \end{align*} We do so on the one hand by considering the sequence of partial sums of the corresponding formal power series analogously as we did in (2) \begin{align*} \sum_{n=0}^\infty b_n(x):=\left(\sum_{n=0}^Nb_n(x)\right)_{N\geq 0} \end{align*} This sequence of partial sums of formal power series may converge to an element $b(x)$ from $\mathbb{K}[[x]]$. In this case we define similarly to (3) the expression (4) as limit of the sequence of partial sums of formal power series. \begin{align*} \sum_{n=0}^{\infty} b_n(x):= \lim_{N\rightarrow \infty}\left(\sum_{n=0}^Nb_n(x)\right)=b(x) \end{align*}

Note that in the professor's text for convenience only $a_N(x)$ is used as shorthand to denote the partial sums \begin{align*} a_N(x):=\sum_{n=0}^Nb_n(x)\qquad\qquad N\geq 0 \end{align*}