For finite series if one wants to be formal is it permissible to use "..." in the course of proofs. What I mean about the question above is if one wants to be formal is it okay to use "..." in a proof for example.
Tao defines finite series like this:
let $m$, $n$ be integers and let $(a_i)_{i=m}^{n}$ be a finite sequence of real numbers, assigning a real number $a_i$ to each integer i between $m$ and $n$ then we can define the finite sum $\sum_{i=m}^n a_i$ by the recursive formula
$\sum_{i=m}^n a_i$ $:=$ $0$ whenever $n < m$
$\sum_{i=m}^{n+1} a_i$ $:=$ ($\sum_{i=m}^{n}a_i$) $+$ $a_{n+1}$ whenever $n\geq m-1$
and then proceeds to say "Because of this, we sometimes express $\sum_{i=m}^n a_i$ less formally as
$\sum_{i=m}^n a_i$ $=$ $a_m$ $+$ $a_{m+1}$ $+$ ... $+$ $a_n$"
Sometimes however it is useful to use $\sum_{i=m}^n a_i$ $=$ $a_{m}$ $+$ $a_{m+1}$ $+$ ... $+$ $a_n$ in proofs. However strictly speaking this is not formal? What about if i prove from the definition above that $\sum_{i=m}^n a_i$ $=$ $a_{m}$ $+$ $a_{m+1}$ $+$ ... $+$ $a_n$ is indeed true (induction trivially obvious)
Also when dealing with double summations it is very helpful to think about finite sums as$\sum_{i=m}^n a_i$ $=$ $a_{m}$ $+$ $a_{m+1}$ $+$ ... $+$ $a_n$ instead of the above recursive definition
 A: You can't "prove" that $\sum_{i=m}^na_i=a_m+a_{m+1}+\cdots +a_n$, because the RHS is not a formally defined thing. What do the "$\dots$" even mean? You could try to tell me "oh it just means you add up all the numbers in the list from $a_m$ up to $a_n$", and I definitely understand what you mean. But if were to really grill you on what exactly you mean, then we have to get down to the nitty gritty details, and you'll admit to me that $\dots$ is informal, because it has no meaning (other than the meaning we infer from context). A computer will not understand "$\dots$" (unless of course you provide it with a specific meaning of $\dots$), but it definitely understands recursion/iteratively doing things. So, the correct way of saying things is to first define the symbol $\sum_{i=m}^na_i$ recursively, and then say that the string of symbols "$a_m+a_{m+1}+\cdots +a_n$" is defined to mean $\sum_{i=m}^na_i$. In this manner, we are being explicit about how the $\cdots$ is to be interpreted.
We humans are sometimes very good at "filling in the blanks". If I were to tell you that I'm thinking of a bunch of numbers according to a certain rule, and I write
\begin{align}
1,2,4,\cdots,64
\end{align}
then any "normal" person would say that the missing numbers are $8,16,32$, and that my "rule" was "multiply the previous number by $2$". However, my "rule" could very well have been "choose a number larger than the previous one", in which case I could have many possible missing numbers. For example the list I'm thinking of could be $\{1,2,4,5,6,7,8,3\pi,64\}$.
The use of $\cdots$ is informal. Sure, we may use it all the time because we're either lazy/ want to quickly convey the meaning/ we are confident that we can rephrase the informal $\cdots$ into a formal way/ we are confident that our reader will not misinterpret things, or any other reason.

Also when dealing with double summations it is very helpful to think about finite sums as$\sum_{i=m}^n a_i$ $=$ $a_{m}$ $+$ $a_{m+1}$ $+$ ... $+$ $a_n$ instead of the above recursive definition.

Well, sure, it may help to "think" about finite sums in that way, but the definition is what it is, and you can't change that. Definitions are meant to remove any ambiguity, which is what the recursive definition does. Once we are familiar/comfortable with the formal definitions, we of course revert to the informal manner of speaking and writing.
As a side remark: I don't "think" of $\sum_{i=m}^na_i$ using the recursive definition. I "think" of it as I'm sure everyone else does: just add up all the numbers in the list. Also, I work with $\sum$ using a bunch of rules, like associativity, what happens when we scalar multiply, how to re-order a double sum etc. These are all things which I've proven once in my life when I first learnt the formal definitions, but never again. We don't always think at the level of definitions. We "work" at a higher lever of the theorems/ideas/analogies etc.

This is just as in English. In school, we learn grammar, punctuation, spelling, etc and all our essays are required to be written formally. However, if we're on the street conversing with a store clerk I doubt any of us speak with perfect grammar and stuff. Same goes for my writing here; it's definitely not perfect, but it gets the meaning across (I hope).
