Gadianton wrote:hmm. wondering if this is good. You know some apologist out there must be considering that if you add up all the numbers from alpha to omega, it sums up to Jesus presiding over the twelve apostles.
ha ha
Well, maybe we should stick with the idea that the asserted equality is a signal that one has used a way of extracting a well motivated answer from a divergent series. It has the advantage of being extremely natural within the context of analytic continuation and also apparently gives physically correct results in certain contexts.
Let me try again to give a feel for it:
Larry: Hey Steven, my analysis of our physical theory has led me to the following series:
1+x^2+x^4+x^6+....
Steven: OK, Great. Is there a problem?
Larry: Yes, I need to plug in the number 2. But then I get 1+4+16+64+.... which doesn't converge.
Steven: I see. Well, doesn't your series converge to 1/(1-x^2) as long as x is between -1 and 1?
Larry: Yes but I need to plug in 2?
Steven: Why not just directly use 1/(1-x^2)?
Larry: Hmmm, well that gives me -1/3 which strangely is just what I expected based on other physical notions.
Steven: Great!
Larry: Hey maybe we could say that in some sense 1+4+16+64+....really equals -1/3.
Steven: Ha ha. Yes, that's a nice way of talking. Did you know Euler actually wrote it down as an equality?
Larry: Oh well, that's interesting.
Steven: Hey Larry, I have my own physical theory about quantum field theory on curved spacetime and it turns out that I also get a divergent series. I just realized that my series is connected to a Zeta function. I wonder if a similar idea might help me there.....