I actually know a bit about summing series, so I recognize the proof as completely bogus but the actual sum as probably correct, for a certain sense of “sum”. You can make a divergent series add up to anything at all by grouping and rearranging terms. On the other hand, there actually are techniques for finding the sum of a convergent series that sometimes don’t give nonsensical answers when you try to use them to find the “sum” of a divergent series, and in this sense the sum of 1 + 2 + 3 + etc. actually can be said to equal −1/12.
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 actually know a bit about summing series, so I recognize the proof as completely bogus but the actual sum as probably correct, for a certain sense of “sum”. You can make a divergent series add up to anything at all by grouping and rearranging terms. On the other hand, there actually are techniques for finding the sum of a convergent series that sometimes don’t give nonsensical answers when you try to use them to find the “sum” of a divergent series, and in this sense the sum of 1 + 2 + 3 + etc. actually can be said to equal −1/12.
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.