functions - Is every sum to something a limit of the sum?

Old title:

Is it always true that a sum to some $m$ equals the limit of that sum to some $x$ such that $x$ tends to $m$?
$\ $

This may seem a stupid question, but I'm placing it because although it seems obviously true, it may not be so in some edge cases. After all, all true statements in mathematics are obvious (or at least analytic and/or tautological propositions).

Also, sorry for the long question title, but I didn't know how to phrase it without using "this".

$$

\forall f,\ \forall n\in\mathbb{N}\cup \{0\},\ \forall m\in\mathbb{N}\cup\{0; \ \infty\},\ \forall x \in D_f, \sum_n^mf(x)=\lim_{x\to m}\sum_{n}^{x}f(x)
$$

Notice that often, $x = n$, they can be the very same variable. So, is the above true (under any circumstances, except wrongness, of course)?

Added: Note: I prefer not to state much about $x$ because even being an upper limit of a sum, it may be non-integer by analytic extension (I don't know if this is the correct name, but I think you get the idea) of the operation, in the limit, as $x$ approaches $m$, it may be a fractional number, perhaps requiring you to (if such is the function) do a half and then to do two thirds of a derivative, and add that to the previous complete derivatives, studying how the partial sums approach the final value.

In the following examples the stated above applies.

Where $x=n$

$$
\sum_{n=0}^{\infty}\left(9\times 10^{-n}\right)=\lim_{x\to \infty}\sum_{n=0}^{x}\left(9\times 10^{-n}\right)=10
$$

Where $x\not=n$

$$
\sum_{n=0}^5 3=\lim_{x\to 5}\sum_{n=0}^{x}3=5\times 3=15
$$

Thank you in advance.

I have found doubtful cases and reminded of this question, such as the one shown here.

It seems quite obvious from evaluating partial sums that

$$
\lim_{n\to\infty}\sum_{i=0}^n2^n=\infty
$$

however:

$$

\sum_{i=0}^\infty2^n=2^0+\sum_{i=1}^\infty2^n=1+2\sum_{i=0}^\infty2^n
\iff\\
\iff-\sum_{i=0}^\infty2^n=1\iff\sum_{i=0}^\infty2^n=-1\neq\lim_{n\to\infty}\sum_{i=0}^n2^n
$$

Which seems to answer "no" to my question. Unless I'm missing something. Am I? Is this logic correct?