This was a fun read and felt (for me) more simple and follow-able than most things I’ve read explaining math! Thank you.
I got up to the sums of powers of 2 being −1. That bit took a few close reads but I followed that there was a pattern where infinite sums of (1,r,r2,r3,…) equal 11−r, and there’s reason to believe this holds for r between −1 and 1. Then you write that if we apply it to r=2 then it’s equal to −1, which is a daring question to even ask and also a curious answer! But what justification do you have for thinking that equation holds for r that aren’t between −1 and 1? I think that this was skipped over (though I may be missing something simple).
(Also, if it does hold for numbers that aren’t between −1 and 1 that I believe this also implies that all infinite sums of rn equal to negative numbers, and suggests maybe all infinite sums of positive integers will too.)
But what justification do you have for thinking that equation holds
I don’t have one: this is simply filling in the the pattern. You can say “No, for any r>1, there is no answer,” (just like you could reasonably say things like ”−1 has no square root”) but if you decide to extend the domain of the function C, this is not just a possible value, it is the value. (The proof of this claim requires actual rigor, but if you’re willing to go along with pattern matching, you get the right answer.)
This was a fun read and felt (for me) more simple and follow-able than most things I’ve read explaining math! Thank you.
I got up to the sums of powers of 2 being −1. That bit took a few close reads but I followed that there was a pattern where infinite sums of (1,r,r2,r3,…) equal 11−r, and there’s reason to believe this holds for r between −1 and 1. Then you write that if we apply it to r=2 then it’s equal to −1, which is a daring question to even ask and also a curious answer! But what justification do you have for thinking that equation holds for r that aren’t between −1 and 1? I think that this was skipped over (though I may be missing something simple).
(Also, if it does hold for numbers that aren’t between −1 and 1 that I believe this also implies that all infinite sums of rn equal to negative numbers, and suggests maybe all infinite sums of positive integers will too.)
I don’t have one: this is simply filling in the the pattern. You can say “No, for any r>1, there is no answer,” (just like you could reasonably say things like ”−1 has no square root”) but if you decide to extend the domain of the function C, this is not just a possible value, it is the value. (The proof of this claim requires actual rigor, but if you’re willing to go along with pattern matching, you get the right answer.)