I know about Cesaro and Abel summation and vaguely understand analytic continuation and regularization techniques for deriving results from divergent series. And.. I strongly disagree with that last sentence. As, well, explained with this post, I think statements like “1+2+3+...=-1/12” are criminally deceptive.
Valid statements that eliminate the confusion are things like “1+2+3...=-1/12+O(infinity)”, or “analytic_continuation(1+2+3+)=-1/12“, or “1#2#3=-1/12”, where # is a different operation that implies “addition with analytic continuation”, or “1+2+3 # −1/12”, where # is like = but implies analytic continuation. Or, for other series, “1-2+3-4… #1/4” where # means “equality with Abel summation”.
The massive abuse of notation in “1+2+3..=-1/12” combined with mathematicians telling the public “oh yeah isn’t that crazy but it’s totally true” basically amounts to gaslighting everyone about what arithmetic does and should be strongly discouraged.
I know about Cesaro and Abel summation and vaguely understand analytic continuation and regularization techniques for deriving results from divergent series. And.. I strongly disagree with that last sentence. As, well, explained with this post, I think statements like “1+2+3+...=-1/12” are criminally deceptive.
Valid statements that eliminate the confusion are things like “1+2+3...=-1/12+O(infinity)”, or “analytic_continuation(1+2+3+)=-1/12“, or “1#2#3=-1/12”, where # is a different operation that implies “addition with analytic continuation”, or “1+2+3 # −1/12”, where # is like = but implies analytic continuation. Or, for other series, “1-2+3-4… #1/4” where # means “equality with Abel summation”.
The massive abuse of notation in “1+2+3..=-1/12” combined with mathematicians telling the public “oh yeah isn’t that crazy but it’s totally true” basically amounts to gaslighting everyone about what arithmetic does and should be strongly discouraged.