… So (1+1+1+\cdots = \infty = 2+2+2+\cdots = 2\times\infty)?
But (\infty -1 \neq \infty)?
It is charitable to assume that I’m capable of understanding, but uncharitable to assume my interlocutors have made their meaning clear.
I don’t know. I agree it seems intuitive, but I think that’s deceptive. By associativity, (1+1+1+\cdots = (1+1)+(1+1)+\cdots = 2+2+2+\cdots )
(= 1+(1+1)+(1+1+1)+\cdots= 1+2+3+4+5+\cdots) And I have it on good authority that that last expression is equal to (-\frac{1}{12}), but I don’t know why.
And ((1+1+1+\cdots)-(1+1+1+\cdots)) could mean taking each term in the one on the right and subtracting it from the one on the left. But normal algebraic manipulation also permits a pattern of (1+1-1+1+1-1+1+1-1+\cdots).
I also take your point that this has implications for (9.\dot9-9), but it is noteworthy that this doesn’t have similar problems. There’s no way to change the outcome by mixing different parts of the series, because the parts don’t overlap: (9 + .9 + .09 -.9 + .009 + .0009 - .09 + \cdots) That difference seems significant, what do you make of it? I have a suspicion that this is related to the divergent/convergent distinction (e.g. that the formal way of saying that is true for convergent series and not for divergent series; that is a weakly held belief, like 65% confidence).
I’m saying that it’s meaningless in the sense of not pointing to a coherent concept. Another example that doesn’t use a logical contradiction would be “the number of dreams it takes to power a Saturn V rocket”. All the words are meaningful, but they don’t point to a coherent concept.
Right, but the idea and the expression are distinct. That you can use an infinite series to express the idea does not make the idea infinite. Or, if we want to insist that it does, then it’s true of any number that can be expressed as an infinite series, which I’m pretty sure is any number. And so in that case there is no distinction being made.
This is not my understanding of that proof.
To give an analogy for the argument I understand you to be making: the number 2 can be expressed as the product of two complex numbers (-2i\times i). So you point out that (0.\dot9) can be expressed as an infinite series, and so you treat any math using (0.\dot9) as “do[ing] arithmetic with covergent series”, but I wouldn’t call the expression (2+2), “do[ing] arithmetic with [complex numbers]”.
Am I to understand that you don’t think any convergent series equals an integer? Otherwise this is completely breaking of like all of algebra.