Let $\mathbb{R}[[X]]$ be the ring of the formal power series. I want to know more about structure of $\mathbb{R}[[X]]$ and the convergences of some known infinite sums in it. My question here is related to the convergence of rational function and infinite sums in the titled ring, for example we have this sequence $(1+\frac{X}{n})^n$ doesn't converges in $\mathbb{R}[[X]]$ however it's converges in $\mathbb{R}[X]$ the ring of polynomials, then my question here is:
Question: What is the difference between convergence in $\mathbb{R}[[X]]$ and $\mathbb{R}[X]$ ?
I'd rather not use $\Bbb X$ as the variable, so if you don't mind, I'll stick with just $x$. Also, for convenience, denote the set of real numbers endowed with the discrete topology by $\Bbb R_d$.
The elements of $\Bbb R[[x]]$ may be identified with their sequence of coefficients: $$\sum_{k=0}^m a^{(k)}x^k \equiv \{a^{(k)}\}_{k\in \Bbb N}$$ where $a^{(k)} = 0$ for $k > m$. (Since we will be discussing sequences of polynomials, I'm reserving subscripts for the sequence elements. Thus the odd notation for term degree.)
By this identification, we can consider $$\Bbb R[[x]] \subset \prod_{n \in \Bbb N} \Bbb R_d = \Bbb R_d^{\Bbb N}$$ under the product topology. Note first of all that the identification is by the coefficients of the polynomials. Thus convergence in $\Bbb R[[x]]$ is about convergence of the coefficients, not convergence for values of the variable $x$. Some other points:
Put these together, and you get the following characterization of convergence in $\Bbb R[[x]]$.
Lemma: A sequence $\{p_n\} \subset \Bbb R[[x]]$ converges if and only if
At first glance, this might appear strong enough to prove pointwise convergence in $x$, but this is not the case. For example, $\{x^n\} \to 0$ in $\Bbb R[[x]]$, but it only converges in $\Bbb R$ for $|x| < 1$.
What you can show is that any convergent sequence $\{p_n\} \subset \Bbb R[[x]]$ can be decomposed into two other sequences $p_n = q_n + r_n$ where $q_n$ is eventually constant and the lowest degree of the non-zero terms in $r_n$ rises without bound (and so $r_n \to 0$).
As for $\Bbb R[x]$, this notation is usually used without regard to topology, so I am not personally aware of any canonical topology on it. There are numerous topologies that can be used. For example, it is a subset of all the $L_p(\Bbb R)$ spaces. But for now, consider it a subspace of $\Bbb R^{\Bbb R}$, once again with the product topology (this time with the usual topology on $\Bbb R$). So convergence in $\Bbb R[x]$ is pointwise in the variable $x$.
Unfortunately, your sample sequence $(1 +\frac xn)^n$ does not converge in either $\Bbb R[[x]]$ or $\Bbb R[x]$. In $\Bbb R[[x]]$ it does not converge because its sequences of coefficients are not eventually constant. In $\Bbb R^{\Bbb R}$, it converges to $e^x$. But $e^x \notin \Bbb R[x]$, so it doesn't converge in $\Bbb R[x]$.
However, it is easy to construct examples of convergent sequences in $\Bbb R[x]$ that do not converge in $\Bbb R[[x]]$: just bound the degrees of the polynomials, and choose convergent sequences of coefficients that are not eventually constant. One such example is $\{(1 + \frac1n)x\}_n$.
Therefore each of $\Bbb R[x]$ and $\Bbb R[[x]]$ have sequences that converge for them but not for the other. Sequences that converge for both include eventually constant sequences. But it also includes sequences like $r_n$ above, provided that the coefficients of the non-zero terms in $r_n$ also decrease in such a way as to allow pointwise convergence. For example, the sequence $\left\{\left(\frac x{2^n}\right)^n\right\}_n$ converges in both without being eventually constant.