I tried to read the Wiki article on this but, I am still mystified at what is happening here.
If we add up an infinite series of integers (1+2+3+4+5+6…) we get an answer of -1/12?
Seems impossible but I have no doubt it is me missing something. So, what am I missing?
(If possible, explain like I am five)
For an infinite summation to make sense, it needs to converge, that is adds up to a specific finite number. I won’t go into the weeds about absolute vs. conditional convergence but suffice it to say that if you have an infinite summation that doesn’t converge you can do crazy things with it.
The overall concept at work here is that when working with infinity, things can happen mathematically that don’t happen when working with finite numbers. I would advise you to watch Mathologer’s video on this
Well, I’ll give you credit for more abstract reasoning than a five year old can be reasonably expected to handle, but here goes:
What we commonly call a “series” is typically a sum of a group of numbers. We most often speak of infinite series (summing an infinite number of terms). For example, 1+1/2 + 1/3 + 1/4 + 1/5 + … Or 1+2+3+4+5+6+…
We can compute what are called “partial sums” of the series, i.e. the sequence we get by taking the sum of the first 1, 2, 3, etc terms of the series.
For 1 + 1/2 + 1/4 + 1/8 + …, the sequence of partial sums would be: 1, 1.5, 1.75, 1.875, and so on.
We say a series is “convergent” if the sequence of partial sums converges to some value. In the case above (1 + 1/2 + 1/4 + 1/8 + …), the series converges to 2.
We say a series is “divergent” is the sequence of partial sums does not converge, i.e. diverges.
A series can diverge in different ways. One way is by becoming unbounded, i.e. going infinite.
For example, 1+2+3+4+5+… is a “divergent series” under normal definitions.
Another example: 1 - 1 + 1 - 1 + 1 - 1 + … The partial sums are 1, 0, 1, 0, 1, 0, 1, 0, … This sequence never really settles down to a particular value so the corresponding series is also divergent.
Ramanujan summation is a different sort of operation that assigns values to divergent series. It can be though of as using some properties of integration (which itself can be thought as a summation process) to define meaningful values for such series.
So it’s not a “summation” in the normal sense we think of in arithmetic but still useful in some contexts.
So, maybe it’s better to think of it as an operation that has analogues to summation. Kind of like how we speak of “imaginary” numbers. Those numbers are not flights of fancy but really do exist, but the connotation with “imaginary” makes people think they aren’t actual numbers. Likewise, calling this a “summation” adds a connotation that we’re discussing arithmetic summation, when we’re really speaking about a different operation.
How many numbers need to be added before the -1/12 answer appears (or becomes apparent)? Or are the numbers converging on -1/12 the further you go?
(I understand the answers above of why 1+2+3+4+5… is not convergent.)
I found this article and the section “heuristics” to be approachable. The OP probably will too. Though you definitely need to start from the top. Which I why I linked to the whole article, not that section.
The main wiki article on Ramanujan summation read to me like “Blah blah Ginger. Blah blah blah”. My undergraduate minor was abstract math. I recognize a few words, but the rest is gibberish. That was 45+ years ago, with little to no use in between. And I’m much lazier than I was.
It is not a convergent series at all: the sum does not become close to -1/12 no matter how many terms you add together. “Ramanujan summation” AIUI starts with the Euler–Maclaurin sum formula.
Well, before it is even relevant to look at any details, the question is how (and why) is it even reasonable to define the sum of a divergent series.
I suppose maybe a better way for the OP to look at this is the Ramanujan summation is not addition. Not at all. As others far more learned than I have already said.
What it is instead is a mathematical mechanism to feed the whole infinite series in at one go (not incrementally as with ordinary addition done in a loop) and derive a single unambiguous (real?) number as a summary (my word) result. In that way different infinite series can be talked about using their “summaries” as stand-ins for themselves. And the ways in which different series’ “summaries” are similar or different gives insight into those series.
What those insights might be is beyond mere me. But it’s meaningful to somebody.
Please understand I meant that entirely as a criticism / admission of my own lack of mathematical skill. Not as a criticism of the article nor of the rationale behind wanting to define a summation operator on infinite diverging series.
I may have given the wrong impression there. If so, sorry to mislead.
As @LSLGuy put very well, it doesn’t happen at all.
Ramanujan summation is not summation in the sense we usually mean. So the partial sums never, ever converge to any particular value.
As above, Ramanujan summation is really an entirely different mechanism where you assign a value to the series as a whole. The mechanics of which can get fairly involved, especially for the lay-person, but not necessary for a high-level understanding.
Think of it as closer to an operation. R(My Series) = some value. But you only assign a value for the infinite series as a whole, not for the partial sums along the way.
If you would rather, let’s invent a different operation. The Antibob number for a divergent series is defined as Antibob(any divergent series) = 1. So, the Antibob summation of 1+2+3+4+… = 1. In this case, this summation is not very useful, but it illustrates the point. If this operation was broadly accepted and useful, then we could write a Wiki page about about Antibob summation as well.
I do not remember exactly what Ramanujan was trying to do, but part of it is to derive various identities (including for infinite series); his theory of divergent series was often a means to an end.
Useful link:
https://www.sussex.ac.uk/webteam/gateway/file.php?name=ramanujans-notebooks.pdf&site=454
[see page 133]
And I think that is ultimately what is going on here as the Mathologer video points out. The whole syllogism can be broken down as
IF those series given to derive Ramanujan summation are convergent THEN 1+2+3+4+… = -1/12. Here is where most people stop.
The second premise is Those series do not converge.
Therefore 1+2+3+4+… = -1/12 may not = -1/12. Note this is not the denying the antecedent fallacy as the sum may or may not = -1/12 but we cannot say for sure given just the methods used to derive it. Looking at the partial sums we can clearly see that it is not -1/12
Here’s a question: Ramanujan had no formal education in mathematics and so I doubt he learned about the subtleties of convergence/divergence. I certainly didn’t until my BA in mathematics. Is it possible that this was his way of showing the paradoxes associated with infinite series being unfamiliar with Reimann’s work? IIRC he never actually believed his statement about 1+2+3+… = -1/12 was true.
Of course, we’re making this a bit more complex than it really needs to be.
How’s this for some mathematical trickery (about this particular case, not Ramanujan summation in general):
We’ll use 2 of the series mentioned above:
S1 = 1 - 1 + 1 - 1 + 1 -1 + …
1 - S1 = 1 - (1 - 1 + 1 - 1 + …)
1 - S1 = 1 - 1 + 1 - 1 + 1 - 1 + …
Does that right side look familiar? It’s S1 again!
1 - S1 = S1
So S1 = 1/2
We know this series never actually gets to 1/2 (it just bounces between 0 and 1), but infinite sums do weird things.
Ok, so let’s add another weird series:
S2 = 1 - 2 + 3 - 4 + 5 - 6 + …
S1 - S2 = (1 - 1 + 1 -1 + …) - (1 -2 + 3 - 4 + 5 - 6 + …)
Regrouping the right side:
S1 - S2 = (1 - 1) + (-1 + 2) + (1 - 3) + (-1 + 4) + …
S1 - S2 = 0 + 1 - 2 + 3 - 4 + 5 - 6 + …
The right side is just S2 again!
S1 - S2 = S2
And since we know S1 = 1/2, that means S2 = 1/4:
S1 - S2 = S2
(1/2) - S2 = S2
S2 = 1/4
Ok, so now to the main show with our original series:
A = 1 + 2 + 3 + 4 + …
S2 - A = (1 - 2 + 3 - 4 + …) - (1 + 2 + 3 + 4+ …)
Again, regrouping terms:
S2 - A = (1 - 1) - (2 + 2) + (3 - 3) - (4 + 4) + …
S2 - A = 0 - 4 + 0 - 8 + 0 - 12 + …
Eliminating the 0’s:
S2 - A = -4 - 8 - 12 - 16 - …
S2 - A = -4*(1 + 2 + 3 + 4 + 5 + …)
S2 - A = -4*A
Since S2 = 1/4:
S2 - A = -4A
S2 = -3A
1/4 = -3*A
A = -1/12
Nifty, eh? If you treat the entire series as a single “thing”, you can do some funky things because of the infinities floating around. But arithmetic involving an infinite number of terms is not as straightforward as all that, hence partial sums and so forth. Even the idea of “pairing up” like terms in infinite series is questionable but the above derivation relies on it.
Ramanujan did have a formal education in mathematics after Hardy got him to England. And even in his informal days, he had a good grounding of the concepts and was ahead of many if not most of the experts of the day.
He did not literally believe he had a summation. He was playing with the concept of divergent series and what you could do with them.
FWIW I got onto this question for the SDMB because I watched a video that tried to show Ramanujan’s genius. The video did this by telling us he had no formal education in math (beyond the basics) and started doing math on his own. He sent a letter to GH Hardy, perhaps the best mathematician of the day (or up there), where, among other things, Ramanujan described this summation.
Make of it what you will. It was enough that Hardy saw a spark of genius and invited him to Great Britain to work further.
Missed edit:
Jump to timestamp 7:53 in video above for the summation.
Yes, Ramanujan was well aware of the difference between convergent and divergent series, if that is what you are asking.
I get that. My question is did he recreate the work of Reimann in that you cannot treat divergent series like a convergent series (or you get nonsensical results) without knowing Reimann’s work?
I believe he did. I cannot say for certain. I recall reading something many years ago (sorry, no cite) that said when Hardy got Ramanujan’s letter he almost cast it aside because so much of the math in there was stuff that had already been done and nothing (or not much) in it was particularly new. But Hardy then realized that Ramanujan had come up with all this math on his own and with no outside help. That suggested he was a math genius so Hardy invited him to England.
Whether the divergent series stuff was wholly invented in Ramanujan’s head or he saw something in a book I do not know.
Ramanujan pretty much knew what he was doing, but it is true that he had some non-rigorous derivations and even incorrect assertions in his notebooks. Not sure what you specifically mean by “treat divergent series like a convergent series”, but see Chapter 6 in the link I posted.
A(n absolutely) convergent series can be manipulated like a finite series while a divergent series cannot. The result from the OP shows what happens if you try.