="James Propp" <jamespropp@gmail.com> Here's a short path from divergent series summation mumbo-jumbo to total nonsense:
Very nice! A fast cutoff from mumbo-jumbo to total nonsense will save me lots of time! (Are we there yet?<;-)
... you'll get a non-real complex number a+bi with b non-zero.
Assuming of course you legitimize imaginary quantities! But of course that was also once controversial. One could instead take the position that this is an argument against this evaluation of the sum, much like:
="Fred Lunnon" <fred.lunnon@gmail.com> ... = 1/(1 - x)^2 at x = 1 , yielding (some flavour of) infinity.
If we can disallow expressions like 1/0 why can't we also disallow a+bi?
="James Propp" <jamespropp@gmail.com> Now take the complex conjugate of the equation 1+1+2+5+...=a+bi.
This assumes that you can take the complex conjugate of this infinite aggregate object by conjugating its components. Since this led to a contradictory result, one could instead argue that you've just proved that this is an inadmissible operation. Can you construct a contradiction from a divergent sum whose generating function at unity is finite and real-valued?
="Dan Asimov" <dasimov@earthlink.net> Yeah, it¹s as simple as that. The definition of the existence and, if so, of the value of an infinite sum is very well-established in math. If you mean something different, how hard is it to say as much?
Well, well-established for some t > Euler anyway. But I agree, math is a game and if you are going to play by unorthodox rules it's more sporting to say so at the outset. Actually I don't mind these textbook arguments as demonstrations that inconsistent rule sets lead to contradictions, thereby highlighting the underlying inconsistencies. I only object to dismissing out-of-hand anything but the most rigidly orthodox interpretation of what the symbol "1+2+3+4+..." means as The One Truth without likewise some acknowledgement of their actually being an underlying adopted canon. Mostly in practice even this doesn't matter, except when the pedagogic purpose is to delight and astonish with unexpected consequences, as it was here, in order to motivate deeper engagement. It's kill-joy to prohibit magicians from positing that rabbits may be extracted from hats.
="James Propp" <jamespropp@gmail.com> Therefore (for the usual reason) I am the Pope.
="Fred Lunnon" <fred.lunnon@gmail.com> Anyway, Jim isn't the pope --- I am.
No need to fight guys, let's just get rid of this pesky "excluded middle"! "To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true; so that he who says of anything that it is, or that it is not, will say either what is true or what is false." --Aristotle, Metaphysics, Book IV, Part 7 (per Wikipedia) Clear as mud! --MLB