Whoops, I oscillated between including the generalization necessary for Chronos’s example and not. Let me clarify:
There are three generalizations in play: I’ll call one overflow-summation (it lets us sum 1 + 2 + 4 + 8 + … = -1, among other things) and the other zeta-summation (it lets us sum 1 + 2 + 3 + 4 + … = -1/12, among other things). There’s also Abel summability, which is weaker than both of those.
When I wrote this, I wasn’t thinking of overflow-summation at all. What I wrote describes Abel summability; the zeta-summable series which are shiftable are precisely the ones which are furthermore Abel summable. If we toss overflow-summation on as well, then this is still a sufficient condition for shiftability, but not necessary. (For example, 1 + 2 + 4 + 8 + … does not meet this criterion, but can be taken as shiftable if one likes; any series which is overflow-summable is also shiftable.).
Again, this is true for zeta-summation. If we toss overflow-summation on as well, then we can allow ourselves to shift some series of positive terms summing to a negative value.
The criteria in terms of cn[sup]p[/sup]b[sup]n[/sup] remains true, even if we use both zeta- and overflow-summation together: we have shiftability just in case there is no component using b = 1.
Here, let me illustrate a taxonomy of summation methods relevant to the discussion, using the geometric series 1 + b + b[sup]2[/sup] + b[sup]3[/sup] + … as a guiding example:
Absolute convergence: These are the nicest sums there are; a series is absolutely convergent to S if, no matter what order you add up its terms in (even things like “First take half of the third term, then the first term, then the other half of the third term, then the second term, …”), you approach S in the limit as you go on. Absolutely convergent summation is invariant under re-ordering, invariant under shifting, and will never turn a series of positives into a negative. It basically has every nice property you can think of. The geometric series will be absolutely convergent just in case |b| < 1. Every summation method below extends absolute convergence.
Traditional summation: This is what you have pounded in your head at school; to take the traditional sum of a series, one one imposes a cutoff point before which every term is brought in with full strength and after which every term is brought in at zero strength. This produces an absolutely convergent approximation, and the traditional sum is the limit of these absolutely convergent approximations as the cutoff point is moved so each term approaches full strength. Traditional summation is not invariant under re-ordering, but is invariant under shifting, and will never turn a series of positives into a negative. In terms of geometric series, traditional summation adds nothing new to absolute convergence.
Abel summation: This rectifies the discrete cutoff problems of traditional summation; to take the Abel sum of a series, one brings in its terms with exponentially decaying strength (actually, the exponentiality doesn’t matter, and we would get the same results using any sufficiently smooth decay function, but I’ll leave that discussion for later…). This often produces an absolutely convergent approximation, and the Abel sum is the limit of these absolutely convergent approximations as the decay rate is lessened so each term approaches full strength. Abel summation is not invariant under re-ordering, but is invariant under shifting, and will never turn a series of positives into a negative. In terms of geometric series, Abel summation adds summability in the case where |b| = 1 but b is not 1. Abel summation extends traditional summation, and every summation method below extends Abel summation.
4: “Extra-Abel” summation: It may be that the approximations used in Abel summation are absolutely convergent for quick decay rates, but not for slow decay rates, so that one can’t take the limit as the decay rate approaches zero. But the function giving the value of the approximations in terms of the decay rate may be smoothly extendible with finite values all the way from its behavior at quick decay rates through slower decay rates up to a value at no decay, giving us what I’ll call the “Extra-Abel” sum. Extra-Abel summation is not invariant under re-ordering, is invariant under shifting, and will never turn a positive into a negative. In terms of geometric series, Extra-Abel summation adds summability in the case where |b| > 1 but b is not > 1.
“Overflow” summation: It may be that the smooth extension used in Extra-Abel summation starts to blow up to infinity at some decay rate before zero, in which case, the Extra-Abel summation as I am using the term will not be defined. But it may be that the blow up is only because our smooth function is the ratio of two other smooth functions, and these other smooth functions extend all the way to a well-defined ratio at the decay rate of zero (in jargon, we can switch from using “analytic” extension to “meromorphic” extension). The value so obtained will be the overflow sum. Overflow summation obviously extends Extra-Abel summation. Overflow summation is not invariant under re-ordering, is invariant under shifting, and may turn positives into a negative. In terms of geometric series, overflow summation adds summability in the case where |b| > 1, unreservedly.
“Zeta” summation: Going back to Abel summation, it may be that the approximations used in Abel summation are absolutely convergent for all nonzero decay rates, but that as the decay rate approaches zero, these approximations blow up towards infinity. However, the function giving the value of the approximations in terms of the (logarithmic) decay rate may still have a finite degree zero term at zero decay, giving us the Zeta sum. Zeta summation is not invariant under re-ordering, is not invariant under shifting, and may turn positives into a negative. In terms of geometric series, zeta summation adds (to Abel summation) summability in the case where b = 1.
SDMB summation: We can combine both the ideas of overflow summation and zeta summation, allowing ourselves to extend the approximation function of Abel summation to a value at zero decay using both meromorphic extension and degree zero term extraction. The resulting summation method consistently, systematically, rigorously handles everything we’ve discussed in these threads. This summation method is not invariant under re-ordering, not invariant under shifting, and may turn positives into a negative. In terms of geometric series, we will have summability for all b.
Hell, Leibniz already observed the sense in which 1 - 1 + 1 - 1 + … = 1/2 (by an argument essentially the same as used in this thread) in 1674! Euler wrote “On Divergent Series” in 1760 (discussing, among other things, 1 + 2 + 4 + 8 + … = -1, by essentially the same technique of “overflow summation” noted here)!
Oh, don’t worry, my impotent outrage wasn’t directed at your edit (which was pretty funny), just the wider world of mathematicians who still act as though these discussions are improper and summation means the thing you learn in school, end of story (and similarly for so many other topics). It’s dispiriting; I often feel about as productive in continuing to note the coherence of “non-traditional” mathematics and the arbitrariness of the curricular canon as I do in discussions on descriptivism vs. prescriptivism.
In both cases they are a “traditional summation”… i.e. it works a same as on a calculator. But try getting -1/12 using your calculator… when you try it just keeps on getting more and more positive and never negative. (though I guess sometimes the calculator can’t do traditional summation - e.g. it can’t handle very large numbers or lots of decimal precision)
> In both cases they are a “traditional summation”… i.e. it works a same as on a
> calculator.
So “traditional” means something that became generally available about the time I entered college. In college when I wanted to balance my checkbook I would have to go over to a lab in the science building and use the Wang calculator there. It was about the size of two shoe boxes. In high school I remember people doing bookkeeping on mechanical calculators. You couldn’t calculate with an arbitrary number of decimal places on those. You had to do your calculations either on integers or on numbers with two decimal places.
My point is that you seem to think that any kind of arithmetic or number system you like is obvious, while any kind less sophisticated than your kind is only used by primitive tribes living in caves and any kind more sophisticated must have been learned from those extraterrestrials who crashed in Roswell in 1947 and will cause any human brain to explode if thought about too long. Look, there are many levels of definitions for numbers and many levels of definitions for arithmetic. You can’t pick a single level and declare that anything else is too far below you or above you to be bothered with. There is a generally accepted definition for numbers and arithmetic that makes sums like 1 + 2 + 3 + 4 + … = -1/12 useful for certain physics problems.
Yes other types of summation besides the traditional one exist but the claim that the sum of natural numbers “equals” -1/12 needs to be explained that it isn’t the traditional sum. That way it explains why the “sum” isn’t positive.
I mean the numberphile video was entitled:
“ASTOUNDING: 1 + 2 + 3 + 4 + 5 + … = -1/12” (their capitalization) then they said “astounding” a few times in the video.
They never said that they weren’t using the traditional summation - even in their “second proof” video:
They even talked about the sum using a calculator but just brushed that aside as if it is impossible that the sum is actually infinite.
I think they are being misleading when they say sum…
maybe it is a little like saying ‘1’ + ‘1’ is not ‘2’!!! (it is ‘11’…) concatentation instead of addition. (or ‘10’ [binary])
They should have known that their audience thinks that “equals” involves the only kind of additional they’re familiar with and don’t know about other methods. I think it is reasonable to assume that when you’re talking about addition it is the traditional one, not one that isn’t even taught in high school (and I didn’t learn it at university either) So thinking that it is positive infinity isn’t incorrect - like numberphile would have us believe.
As far as old calculators go, I already pointed out that calculators have limited decimal precision. Though we’re talking about natural (whole) numbers - it is similar to counting sticks… it doesn’t make sense that the sum would involve negative 1/12 of a stick. Yes it is -1/12 in a way, but that way needs to be mentioned since it doesn’t involve the typical definition of summation.
Yes the video is a bit misleading in that they don’t point out exactly what they’re doing and how a different definition of summation of an infinite series is needed to the standard one people meet when they first encounter analysis, but then again with popularizing media like this you don’t expect a discussion of the technicalities.
However I don’t see in what sense the ‘+’ signs are different: it’s the same binary operation on whatever set we feel is the relevant set whether we’re talking about the summation of convergent or divergent series. The difference is that we’re now taking a different limit from the one which we encounter in basic analysis.
For infinite convergent series we are used to taking the limit of the sequence of partial sums to evaluate the series, but in order to evaluate divergent series a different, not so easy to describe, limit must be taken depending on which summation method is used. It must be pointed out though that when we apply this technique (assuming it is regular) to convergent series we get the same answer as if we had taken the limit of the sequence of partial sums.
Which summation is the traditional summation, the one that acts on integers or the one that acts on real numbers? You said “the” traditional, so I assume you mean there’s only one.
This will not persuade anyone of anything, but I’d just like to note that calculators do whatever they’re designed to do. You could design a calculator to respond to the input “The sum of the series whose nth term is n” with the output “-1/12” if you wanted to; you could even do this in a systematic way rather than ad hoc.
That your desk calculator doesn’t do this is not a mathematically significant fact, just a sociological one.
I remember when I got my first programmable calculator, it had a program that would quickly solve quadratic equations, but it also had a program that would ask you to type in your name and would return “Hello <your name>, you are a wanker”. The lesson here being that we shouldn’t take everything calculators say to heart.
