Something else occurred to me. You might say "Well it’s obvious the sum:
1 + 1/2 + 1/4 + 1/8 + 1/16 + . . .
adds up to 2. Nobody could possibly doubt that."
Well, somebody did doubt that. It’s called Zeno’s Paradox:
Until someone figured out what infinite addition meant, this was a really hard problem. In any case, it’s necessary to make clear definitions in mathematics. Sometimes these will strike you as arbitrary definitions.
This is what I tried to show back in Post #17, but with a slightly different example: 1 + 2 + 4 + 8 + 16 + … = -1.
How could a bunch of positive numbers come to anything negative? It can’t, really. But, here’s the logic:
Step 1: 1 + 2 + 4 + 8 + … can be considered as a particular instance of 1 + x + x[sup]2[/sup] + x[sup]3[/sup] + …, when x is 2.
Step 2: For some choices of x, 1 + x + x[sup]2[/sup] + x[sup]3[/sup] + … has the same value as 1/(1-x).
Step 3: So it makes sense to consider 1 + x + x[sup]2[/sup] + x[sup]3[/sup] + … and 1/(1-x) to be in some sense the same, or to represent the same function of x.
Step 4. 1/(1-x) has a value of -1 when x = 2, and if 1 + x + x[sup]2[/sup] + x[sup]3[/sup] + … is in some sense the same as 1/(1-x), then 1 + x + x[sup]2[/sup] + x[sup]3[/sup] + … would in that sense have a value of -1 when x is 2.
Very well said, Thudlow.
So you have a function which makes intuitive sense on a certain subset of numbers. And there’s a “standard” (smooth) way to extend those types of functions to a larger set of numbers, even if the extension doesn’t make intuitive sense to the layman.
Actually, this strikes me as an astute and valid analogy. Those “joke proofs” show that you can end up with nonsense if you perform algebraic manipulations that are perfectly valid provided the expressions involved are nonzero, but with an expression that is zero—and that’s something that doesn’t just happen in trick proofs but can be a subtle error in serious mathematical work if you’re not careful. Similarly, you can get “nonsense” if you work with infinite series by performing manipulations that would be algebraically valid if the sums were all finite.
An equation like the one in the thread title is silly on it’s own—it’s just sort of a whimsical by-product of serious mathematical study (I hope it’s fair to say).
In a way, it is. That’s why I kept using the weasel words “in some sense” in my example posted above: to avoid making an error by claiming that something’s valid when it’s not.
If someone were to make a statement like this:
Premise a)
If the result of the addition of multiple numbers must be negative, then at least 1 of the numbers being added must be negative
Premise b)
If, in a series of numbers, the first number is not negative and each number beyond the first has a greater value than the first number, then no number in the series is negative
Let T = any set of numbers, which, when added together, result in a negative number
Let Y = any set of numbers, where the first number is not negative and each additional number is greater than the first number
according to premise a, set T must contain at least one negative number
according to premise b, set Y cannot contain any negative numbers
therefore, T does not equal Y
therefore if 1+2+3 fits the definition of set Y, and the sum of a series of numbers equaling negative -1/2 fits the defintion of set T, then the statement that 1+2+3+… = -1/2 cannot be true.
What part of the statement would be found erroneous in order to let the equation be true?
It would seem to me that the obvious error would be:
…
Step 2: For some choices of x, (something) has the same value as (something else)
…
Step 4: if x = 2, then (something) = (something else) = -1
In step 2 you say that, for some choices of x, (something) = (something else), and then in step 4 you say “let’s apply that when x = 2”. But you’ve never proved, in step 2, that (something) = (something else) for all values of x. So it seems likely that the “some values of x” in step 2 does not include “x = 2”.
The analytic continuation of the Riemann Zeta function is not literally Sum(1/n^s,n=1…infinity) for every single real (or complex) number s. It just happens to agree with that formula on a limited subset of numbers.
Thanks for the very clear layman’s explanations, Wendel and Thudlow. Although there’s one part I’m still not clear on:
How do you get from here to 1/(1-x) ?
Indeed. Without the s, it merely leaves us jonesing for a zeta.
Are you asking why 1 + x + x[sup]2[/sup] + x[sup]3[/sup]… = 1/(1 - x) (for |x| < 1 on the standard account of infinite summation; perhaps more generally on other accounts of summation)?
The idea is this:
1 - x[sup]n+1[/sup] = (1 - x)(1 + x + x[sup]2[/sup]… + x[sup]n[/sup]). [Go ahead and multiply the right side out, and you’ll see it comes to the left side]
Thus, (1 - x[sup]n+1[/sup])/(1 - x) = 1 + x + x[sup]2[/sup] … + x[sup]n[/sup]. Letting n go to infinity, and using the fact that x[sup]infinity[/sup] will go to 0 if |x| < 1, we obtain that 1/(1 - x) = 1 + x + x[sup]2[/sup] + …
If the function is not summing
1 + 2 + 3 + …
then isn’t writing it as
1 + 2 + 3 + …
a bad idea?
Beyond this - the statement in my post doesn’t require that one be attempting to add every real or complex number, so I’m not sure how your response deals with the issue. Take another look, maybe?
Yes, that’s exactly right.
Well, the statement isn’t true. (It was actually -1/12, not -1/2, but that doesn’t make it true either.) The only person actually claiming that 1 + 2 + 3 + … does = -1/12 is Ramanujan. I went back and looked at the book the OP originally referred to, to find the context for this equation. (Here’s a link to the context in Google books, if the link works right—if not, try googling “Ramanujan’s papers claiming to have proved that” or get the physical book and turn to p. 135.)
1 + 2 + 3 + … = -1/12 is not something a modern, formally-trained mathematician would write. But Ramanujan was a largely self-taught, untrained mathematical genius when he wrote that and sent it to some of Britain’s leading mathematicians, two of whom were able to figure out what he meant by it and in what sense it actually made sense (which is what this thread has been trying to answer).
I want to note that, in a way, this is the grand-daddy of many of the most famous infinite series; that 1 - 1/2 + 1/3 - 1/4 … = ln(2), that 1 - 1/3 + 1/5 - 1/7 … = π/4, and that 1/1[sup]2[/sup] + 1/2[sup]2[/sup] + 1/3[sup]2[/sup] + … = π[sup]2[/sup]/6 all can be derived starting from 1 + x + x[sup]2[/sup]… = 1/(1 - x), as can the above results that 1 - 1 + 1 - 1 … = 1/2 and that 1 - 2 + 3 - 4 … = 1/4 (though the convergence is only in a suitably averaging sense), and from them that 1 + 1 + 1 + 1 … = -1/2 and 1 + 2 + 3 + 4 … = -1/12 in a further generalized sense.
(Thudlow, Ramanujan isn’t the only mathematician to make note of a sense in which 1 + 2 + 3 + 4 + … = -1/12. Euler did so as well, and apparently even physicists make use of this is renormalization)
Er, “in renormalization”. Also, don’t get the impression that Ramanujan was so sui generis that he didn’t realize that 1 + 2 + 3 + 4 + … is a divergent series on the standard account or note how unconventional it would be to consider it to equal -1/12; in the letter in which he communicated the idea to Hardy, he explicitly added all these disclaimers upfront.
Even modern mathematicians are happy to note the sense in which 1 + 2 + 3 + 4 + … = -1/12; they might choose to write it a little differently from that, but it’s entirely the same idea, regardless of what notation is used to express it.
Yes, it’s extremely misleading. As Thudlow points out, no mathematician would write it that way, except for a joke or an interesting anecdote about Ramanujan.
Ramanujan wasn’t as non-educated or cut off in his youth from the mathematical community as the romantic story would have you think. There were mathematicians (and universities and textbooks and all the rest of it) in India too… Hardy and Littlewood helped bring Ramanujan to international fame, but he had ample access to the same education as any other mathematician even before contacting them.
This is interesting and I found both your posts and Boink’s to be helpful, but I still am of the mind that either:
1+2+3+… does not equal negative anything; or
1+2+3+… does not mean one plus two plus three, etc. in which case writing it as 1+2+3+… is unhelpful at best
I’m not 100% sure which outcome you are implying (the thrid option being that if you add these positive numbers together, then the result is negative - which I do not believe you are stating), but I think perhaps the second.
That makes sense, in some ways - although it seems inappropriate for any book that is attempting to be informative and dealing with an audience that isn’t ‘in on the joke.’
I think one of the most important things to take away from this is that this is not a “proof” that 1+2+3… = -1/12. It is a definition, or rather the consequence of a definition. In a technical sense, those two are almost the same thing (a proof is in some sense a consequence of a definition), but I think what most people consider a proof they think it means that this is some unassailable truth. A more careful statement is that in math, something proven true is proven for under a certain set of definitions and assumptions. For most people, and indeed for much of human history, those assumptions were not well known.
Once they were mostly understood, it became possible to play around with them. A lot of modern mathematics involves changing assumptions, or coming up with new definitions, to see what the consequences are.
For example, I could define addition of two numbers to mean “add half the second number”. Then 2 + 2 = 3, 5 + 8 = 9, 1 + -1 = 1/2. That’s hardly useful except to confuse you. (A particular problem : 1 + 2 = 2, but 2 + 1 = 2 1/2).
Or I could work in a system where 0.9999… is not equal to 1. (Under the ‘standard’ assumptions, that must be true, however). If I do so, I have to redefine a few other things, but it can actually be pretty similar to the familiar system. This actually has had applications as well.
In the case of this summing method, the new definition ‘fits in’ fairly well, and thus also has potential applications. Most commonly a definition that ‘fits in’ like this is going to be the most promising, but really anything novel might eventually work out.
As ultrafilter pointed out, this is not the only way to do this, and probably the other methods have their usefulness as well (otherwise they’d likely be abandoned).
Yes, I could see how 1 + 2 + 3 + … = -1/12
would work when the symbol -1/12 was defined to mean to the sum of the numbers prior to the equal sign 