Ambiguity of some infinite math expressions

I’ve come to realize that I don’t actually know precisely how to interpret some infinite math expressions with an ellipsis (“…”), like

$\sqrt{1 + \sqrt{1 + \sqrt{1 + \dots} } }$

First, to flatten out the syntax, I’ll define $R$ to be the square root function: $R(x) = \sqrt{x}$. Now evaluate the following expressions:

1. $R ( 1 \times R ( 1 \times R ( 1 \times \dots ) ) )$
2. $R ( 0 + R ( 0 + R ( 0 + \dots ) ) )$
3. $R ( 1 - 1 + R ( 1 - 1 + R ( 1 - 1 + \dots ) ) )$
4. $R ( 1 - 1 + R ( 1 - 1 + R ( 1 - \dots ) ) )$
5. $R ( R ( R ( \dots ) ) )$

I get the impression that when a mathematician sees an expression like (1), they mentally interpret it as the limit of an infinite series:

• $R ( 1 )$
• $R ( 1 \times R ( 1 ) )$
• $R ( 1 \times R ( 1 \times R ( 1 ) ) )$
• $\dots$

Which evaluates to 1. They would interpret (2) as the limit of:

• $R ( 0 )$
• $R ( 0 + R ( 0 ) )$
• $R ( 0 + R ( 0 + R ( 0 ) ) )$
• $\dots$

Which evaluates to 0. Yet, with trivial algebraic manipulation, all five of the expressions I listed are identical. It seems odd that they would evaluate differently.

What’s special about expressions like this is that the “…” takes the place of the part that must be evaluated first. There’s no such problem with expressions where the “…” can be evaluated last, like $\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$. Only the most pathological manipulation of this expression could make it evaluate to anything other than 2. The issue I’m discussing only occurs with expressions that are in some sense “back to front”, or “inside out”.

With expressions (1) and (2), there is only one way to abbreviate them by simple deletion of symbols, to make a sequence of partial expressions that are syntactically valid. This seems to be what (some) mathematicians (sometimes) tend to do. But sometimes there is no such way to abbreviate an expression, and this is common enough that I don’t think you can just declare such expressions to be invalid. And, sometimes, there is more than one way to abbreviate an expression, as with (3). What do you do then? Does the placement of the “…” make a difference?

Another way to interpret such an expression is to artificially terminate the sub-expressions by replacing the infinite part with some intuitively-chosen constant, usually 0 (for an addition-oriented expression) or 1 (for a multiplication-oriented expression). E.g., for expression (5):

• $R ( 1 )$
• $R ( R ( 1 ) )$
• $R ( R ( R ( 1 ) ) )$
• $\dots$

So it evaluates to 1. But it does raise the question of why 1 is the proper seed value.

There is a more general way to interpret such an expression, and it’s the way that feels most natural to me. Replace the ellipses with some symbol, say $t$. For expression (5), we construct this sequence:

• $R ( t )$
• $R ( R ( t ) )$
• $R ( R ( R ( t ) ) )$
• $\dots$

The solution set will then be all the possible limits of the sequence, for any possible value of $t$. We would definitely have to know the domain of $t$ (and of the intermediate expressions) — is it real numbers, complex numbers, or what? Of course, that’s often a prerequisite when doing even fairly basic algebra, but in this case it seems particularly obvious that we’re sort of conjuring up something out of nowhere, so we ought to be careful.

And we should probably ignore values of $t$ for which we encounter an expression that cannot be evaluated. An example would be if we have to take the square root of a negative number, and we’re only allowing real numbers. I don’t think it’s necessary to say that one of the solutions is “error”.

Whether “divergent” could be a solution, when convergent solutions also exist, might have to depend on the context of the problem.

With this interpretation, all five expressions have two solutions: 0 and 1. (Whether complex numbers are allowed doesn’t change anything in this case.) I know that some of the expressions might look like they really shouldn’t evaluate to 0, but personally I’m okay with it.

I’ve seen math presenters use all the different interpretations I’ve mentioned above, usually with no explanation for why they chose that interpretation. And I’ve seen arguments about this issue in the comments section of math videos, and that sort of thing, so I know I’m not the only one who thinks it’s not always obvious what to do.