Writing the limit of a series inside the series

Question

Let $f(x) = \sum\limits_{n=0}^{\infty} a_n x^n$ be finite or infinite, where $x$ is a real number and $(a_n)_n$ is an infinite sequence of positive integers. For any integer $n$, let $ (a_{L,n})_{L}$ be an infinite sequence of integers which equals $a_n$ for any $L$ large enough and for any L, let $$ f_L(x) = \sum\limits_{n=0}^{\infty} a_{L,n} x^n. $$ Is it always true that $$ \lim_{L \rightarrow \infty} f_L(x) = f(x)? $$Is there a famous theorem that can be applied from which the identity follows?

Answer 1

I can interpret "$(a_{n,L})$ equals $(a_n)$ for $L$ large enough either as "for every $n$ there exists $k$ such that $a_{n,L}=a_n$ for $L>k$" or as "there exists $k$ such that for $L>k$ and all $n$, $a_{n,L}=a_n$.

In the first case, your statement is false. Let $a_n=1$ and $a_{L,n}=a_n=1$ for $L>n$ and $a_{L,n}=0$ else. Then $\sum a_n x^n=\frac1{1-x}=f(x)$ and $\sum a_{L,n} x^n=\sum_{n=L+1}^\infty x^n = \frac{x^{L+1}}{1-x}=f_L(x)$. For $-1<x<1$, $f_L(x)\to 0\neq f(x)$.

In the second case, your statement is true, as $a_{L,n}=a_n$ for $L>k$ and for all $n$. So for every $L>k$, $f_{L}(x)=f(x)$. Then of course $\lim_{L\to\infty} f_L(x)=f(x)$.

Or do you have another different definition in mind?