The first is simply a shorthand for “the limit of this sum is 2”,
It doesn’t need to be! It can be more generally something like “the unique value that matches patterns,” where what counts is extended first beyond integers, and then to infinite series, and then to divergent infinite series.
You run into the trouble of having to defend why your way to fit the divergent series into a pattern is the right one—other approaches may give different results.
The claim is that they don’t: any pattern that points to a finite result points to the same one. If you want proof, then you need a more rigorous formalism.
Now the problem is this pattern leads to a contradiction because it can equally prove any number you want. So we don’t choose to use it as a definition for an infinite sum.
So you need to do a bit more work here to define what you mean here.
It doesn’t need to be! It can be more generally something like “the unique value that matches patterns,” where what counts is extended first beyond integers, and then to infinite series, and then to divergent infinite series.
You run into the trouble of having to defend why your way to fit the divergent series into a pattern is the right one—other approaches may give different results.
The claim is that they don’t: any pattern that points to a finite result points to the same one. If you want proof, then you need a more rigorous formalism.
Sure, but you’re just claiming that, and I don’t think it’s actually true.
That’s clearly not true in a general sense. Here’s a pattern that points to a different sum:
1 + 2 + 3 + … = 1 + (1 + 1) + (1 + 1 + 1) + … = 1 + 1 + 1 + … = − 1⁄2
Now the problem is this pattern leads to a contradiction because it can equally prove any number you want. So we don’t choose to use it as a definition for an infinite sum.
So you need to do a bit more work here to define what you mean here.