Interesting, I’ve never looked closely at these infinitely-long numbers before.
In the first example, It looks like you’ve described the infinite series 9(1+10+10^2+10^3...), which if you ignore radii of convergence is 9*1/(1-x) evaluated at x=10, giving 9/-9=-1. I assume without checking that this is what Cesaro or Abel summation of that series would give (which is the technical way to get to 1+2+3+4..=-1/12 though I still reject that that’s a fair use of the symbols ‘+’ and ‘=’ without qualification).
Re the second part: interesting. Nothing is immediately coming to mind.
In the first example, It looks like you’ve described the infinite series 9(1+10+10^2+10^3...), which if you ignore radii of convergence is 9*1/(1-x) evaluated at x=10, giving 9/-9=-1.
Yes, this is one way of justifying the claim that −1 is the “right” answer, via analytic continuation of the function 9/(1 - x). But there’s another arguably more fun way involving making rigorous sense of infinite decimals going to the left in general.
I assume without checking that this is what Cesaro or Abel summation of that series would give (which is the technical way to get to 1+2+3+4..=-1/12
Cesaro and Abel summation don’t assign a value to either of these series.
Interesting, I’ve never looked closely at these infinitely-long numbers before.
In the first example, It looks like you’ve described the infinite series 9(1+10+10^2+10^3...), which if you ignore radii of convergence is 9*1/(1-x) evaluated at x=10, giving 9/-9=-1. I assume without checking that this is what Cesaro or Abel summation of that series would give (which is the technical way to get to 1+2+3+4..=-1/12 though I still reject that that’s a fair use of the symbols ‘+’ and ‘=’ without qualification).
Re the second part: interesting. Nothing is immediately coming to mind.
Yes, this is one way of justifying the claim that −1 is the “right” answer, via analytic continuation of the function 9/(1 - x). But there’s another arguably more fun way involving making rigorous sense of infinite decimals going to the left in general.
Cesaro and Abel summation don’t assign a value to either of these series.