Defining ellipsis "$\ldots$" formally in Mathematics for infinite expressions.

Question

My motivation to ask this question stems from this question.

The ellipsis notation is used in two ways:

1. For finite number of terms: When you want to save space and time, you may omit the terms between the first few terms and the last term by using an ellipsis, provided that the few terms have clearly established the pattern/sequence that each of these terms must have.

2. For infinite number of terms: When you have established a pattern/sequence with the first few terms, you use ellipsis after those few terms to mean "and so on" for the rest of the infinite terms.

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I want to discuss particularly the second case.

For example; when you have an infinite sum or product, it is understood that:

• $a_1+a_2 + a_3+ \ldots := \lim\limits_{n \to ∞} \left(a_1+a_2 + a_3+ \ldots+ a_n \right)$
• $a_1 \cdot a_2 \cdot \ldots := \lim\limits_{n \to ∞} \left(a_1 \cdot a_2 \cdot \ldots a_n\right)$

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Now consider another example. If I write down something like this: $$\sqrt{a_1 + \sqrt{a_2 + \sqrt{ \ldots }}}$$ Any reasonable person would interpret and/or define this as: $$ \lim\limits_{n\to∞} \sqrt{a_1 + \sqrt{a_2 + \sqrt{ \ldots +\sqrt{a_n}}}}$$

Or here's one more: $$ f_1\left(f_2\left(f_3\left( \ldots (x) \right) \right) \right):= \lim\limits_{n \to ∞} f_1\left(f_2\left(f_3\left( \ldots f_n(x) \right) \right) \right) $$

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However these are only examples of how one would define or give meaning to an infinite expression which would otherwise be a meaningless garbage.

My question: Is there a way to define "ellipsis" in infinite expressions using limits (or any other means) in general (I only list down examples), to give it a rigourous meaning?

If not, then is it not possible to do something like this for all kinds of infinite expressions? Some counterexamples would help in that case.

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Edit 2: After some thought, what I was sort of trying to ask is if something like this could be done.

Let $(x_n)$ be an infinite sequence of real numbers. We need an infinite sequence to make any sense out of the ellipsis notation, otherwise I assume it would be ambiguous.

Let $E$ be an infinite expression in $x_i$'s for each $i$. Note that, expression is once again, an informal word. But in this context, I would like it to mean "some combination of symbols that have some meaning in $\mathbb{R}$"

So, $E=E(x_1,x_2,x_3, \ldots, x_n, \ldots)$, this is an infinite expression. And to make any sense out of $E$, I'd define $E(x_1,x_2, \ldots):= \lim\limits_{n \to ∞} E(x_1,x_2, \ldots, x_n)$. How's this for a change? Now one could also replace the set $\mathbb{R}$ with some arbitrary set $X$ if they like in which case my sequence would then be in $X$. Does this make any sense or have I just gone complete nuts?

Answer 1

I believe you can eliminate the ellipsis "..." if you use recursive definitions with conditional expressions such as the following.

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$$\sqrt{a_1+\sqrt{a_2+\sqrt{\ldots}}}=\lim\limits_{n\to\infty}\sqrt{a_1+\sqrt{a_2+\sqrt{\ldots+\sqrt{a_n}}}}=\lim\limits_{n\to\infty}b_{1,n}\tag{1}$$

$$b_{i,n}=\sqrt{a_i+\left\{\begin{array}{cc} b_{i+1,n} & i<n \\ 0 & \text{True} \\ \end{array}\right.}=\sqrt{\left\{\begin{array}{cc} a_i+b_{i+1,n} & i<n \\ a_n & \text{True} \\ \end{array}\right.}\tag{2}$$

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$$f_1\left(f_2\left(f_3\left(\ldots(x)\right)\right)\right)=\lim\limits_{n\to\infty}f_1\left(f_2\left(f_3\left(\ldots f_n(x)\right)\right)\right)=\lim\limits_{n\to\infty}g_{1,n}\tag{3}$$

$$g_{i,n}=f_i\left(\left\{\begin{array}{cc} g_{i+1,n} & i<n \\ x & \text{True} \\ \end{array}\right.\right)\tag{4}$$

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Formulas (2) and (4) above can be written in a more standard functional form as follows which is the way they'd actually be implemented in Mathematica.

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$$b(i,n)=\sqrt{a(i)+\left\{\begin{array}{cc} b(i+1,n) & i<n \\ 0 & \text{True} \\ \end{array}\right.}=\sqrt{\left\{\begin{array}{cc} a(i)+b(i+1,n) & i<n \\ a(n) & \text{True} \\ \end{array}\right.}\tag{5}$$

$$g(i,n)=f_i\left(\left\{\begin{array}{cc} g(i+1,n) & i<n \\ x & \text{True} \\ \end{array}\right.\right)\tag{6}$$