How do we know there is any such thing; let alone how to compute it or evaluate it?
The basic axioms of real numbers say that any FINITE sum of real numbers is defined and exists.
If you then give me an expression such as 1/2 + 1/4 + 1/8 + 1/16 + … then what on earth could it possibly mean? The basic rules of the real numbers do not say!
In math we try to define everything from the ground up. There is a formal definition of +, there’s a formal definition of real numbers, and so forth.
So if you see an expression – which has no defined meaning in math – like 1/2 + 1/4 + 1/8 + 1/16 + … ; then how do you know what it means? You have to define it, just like you have to define integrals and derivatives and topological spaces and quantum field theories in physics.
Look at it this way. Suppose I ask you, what is (3 \otimes 47) You can’t tell me. You would have to first ask: How is (\otimes ) defined. Right? I hope you can agree.
It’s the same for 1/2 + 1/4 + 1/8 + 1/16 + … The sum of any FINITE number of terms is defined; but not the sum of infinitely many. We have to define that first. Otherwise we have no idea what it means; or if the notation can even be made logically consistent with the rest of math.
It turns out that it can. By the definition given on the Wiki page. By defining the sum of an infinite series as the limit of the sequence of partial sums, each one of which is finite hence defined, we are defining a NEW notation in terms of things we already know. That’s science! That’s logic.
ps – I wanted to add that I think we’ve arrived at a good place. I see the core issue.
You don’t realize that everything we write down in math must be formally defined and shown to be sensible. We can define the number 3. We can define plus and times. We define everything. Even the things we’ve taken for granted since we were children, must be formally defined once we formally study the subject.
So if we want to talk about infinite sums, we have to define them.
In biology, we can’t just say, “A flying elephant is an elephant that files,” and then open research labs to study them. Since they don’t exist, it’s pointless. Likewise with infinite sums. They don’t have any a priori mathematical existence. We have to define them and show that our definition makes sense in the context of the rest of math.