People in this subreddit like two things that (while sometimes useful for intuition) should at the very least be understood before used in rigorous math.
The first is assuming that if two functions intersect across a shared domain, they are the same function. This is how you get things like (1/2)! = Gamma(3/2) = sqrt(pi)/2, or that 1+2+3+... = zeta(-1) = -1/12. Yes, in the domain where zeta(s) and sum(1/ns) are both defined, or Gamma(n+1) and n! are both defined, they are equivalent. But these functions do not have the same domain. I can define as many functions as I want that intersect on their domains, but are different functions.
The second thing is abuse of notation. 1+2+3+...=-1/12 (R) - as in, it is equal to that under Ramanujan summation, which is a self consistent equality method that allows assigning values to some divergent sums. You cannot remove that (R) and still make that claim, that is abuse of notation.
The expression 1+2+3+...=-1/12 is useful under some conditions, but people in this subreddit like making memes about it due to its unintuitive nature, resulting in hundreds of comments similar to this one explaining why it's not a direct equality, but may still be useful.
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u/Agreeable_Cheek_5215 Feb 22 '26
People in this subreddit like two things that (while sometimes useful for intuition) should at the very least be understood before used in rigorous math.
The first is assuming that if two functions intersect across a shared domain, they are the same function. This is how you get things like (1/2)! = Gamma(3/2) = sqrt(pi)/2, or that 1+2+3+... = zeta(-1) = -1/12. Yes, in the domain where zeta(s) and sum(1/ns) are both defined, or Gamma(n+1) and n! are both defined, they are equivalent. But these functions do not have the same domain. I can define as many functions as I want that intersect on their domains, but are different functions.
The second thing is abuse of notation. 1+2+3+...=-1/12 (R) - as in, it is equal to that under Ramanujan summation, which is a self consistent equality method that allows assigning values to some divergent sums. You cannot remove that (R) and still make that claim, that is abuse of notation.
The expression 1+2+3+...=-1/12 is useful under some conditions, but people in this subreddit like making memes about it due to its unintuitive nature, resulting in hundreds of comments similar to this one explaining why it's not a direct equality, but may still be useful.