I'm studying about formal power series in ring theory from Grillet's Abstract Algebra. I'll go first with some definitions. We don't have a topology here, so we can't use limits, and this is why these definitions are formal.
Def: let $M=\{1,X,\ldots,X^n\}$ be the free monoid on $\{X\}$. A formal power series $A=\sum\limits_{n\geq0}a_nX^n$ in the indeterminate $X$ over a ring $R$ [with identity] is a mapping $A:X^n\to a_n$ of $M$ into $R$. These can be added pointwise and the product is $AB=C\text{ when }c_n=\sum\limits_{i+j=n}a_ib_j~\forall n\geq0.$
Then, if $R$ is a ring, formal power series over $R$ in the indeterminate $X$ constitute a ring $R[[X]]$.
Def: the order ord $A$ of a formal power series $A=\sum\limits_{n\geq0}a_nX^n\neq 0$ is the smallest integer $n\geq 0$ such that $a_n\neq0$.
The next definition is where I'm having trouble.
Def: a sequence $T_0,T_1,\ldots,T_k$ of a formal power series $T_k=\sum\limits_{n\geq0}t_{k,n}X^n\in R[[X]]$ is addible, or summable, in $R[[X]]$ when, for every $n\geq 0$, $T_k$ has order at least $n$ for almost all $k$. Then the sum $S=\sum\limits_{k\geq 0}T_k$ is the power series with coefficients $s_n=\sum\limits_{k\geq0}t_{k,n}$.
If for every $n\geq0$, ord $T_k\geq n$ for almost all $k$, this means that for every $k$ except for a finite amount of indexes, the order of $T_k$ is arbitrarily large? What's the intuition in this? I can't see it very clear nor know how to reason about it. Any clarification or help would be appreciated.
