You did all the hard work:
Calling our sum S = 1 + 2 + 3 + 4 + 5 + 6...
We'll subtract 1/4 = 1 - 2 + 3 - 4 + 5 - 6...
Which gives S - 1/4 = 0 + 4 + 0 + 8 + 0 + 12...
equivalent to S - 1/4 = 4(1 + 2 + 3 + 4...)
equivalent to S - 1/4 = 4S
equivalent to - 1/4 = 3S
equivalent to S = -1/12
To be clear, these are the answers you get when you say “I know it diverges and I don’t care. Give me the sum anyway.”
I didn’t divide by zero or commit any other inexcusable error. The resulting sums have most of the properties that you want, such as linearity. Add two sequences term-by-term and the result is the sum of the two sums. Scale a sequence by some factor A and the sum is scaled by A as well.
The really weird part is that there are some real-world consequences to these results. Without some mathemagical trickery, it appears that the Casimir force should be infinite, since it involves a sum proportional to 1+2+3+4+… Instead it’s small and negative: what you would get if you had plugged in -1/12, in fact. I’m sure that physicists would be much less happy about this if it weren’t that the calculation worked.
This is completely and utterly false. We’ve covered 0.999… = 1 many, many, many, many, many times on this board, and I’m not going to go over it again. Suffice it to say that 0.999… is a number— not a sequence, not a “journey” (whatever that means), or any other hand-waving nonsense— and is equal to 1.
It’s definitely divergent. There are various way to assign “sums” to ostensibly divergent series; ultimately, they’re just maps F from spaces of sequences to R with some some of reasonable conditions imposed: linearity, mapping F(a_n) to \sum a_n if the latter converges, and so on. The most common example is Cesàro summation. There are more exotic forms that basically boil down to throwing some quickly-decaying factor g(n, s) to the form and then removing it by analytic continuation or something similar. (Zeta-function regularization in particular pops in frequently in physics, which is where the hoary result that 1 + 2 + … = -1/12 comes from.) It’s a perfectly legitimate construction, but no one is claiming that the partial sums 1 + … + n approach -1/12. I wish the particular Youtube channel in question had never put up that damn video.
The Casimir effect is actually one of the easier computations to justify: Choose your favorite regulator, churn through the integral, and get a result that only depends on the lowest-order term of the regulator. It even makes sense physically, since we get the same result from imposing an ultraviolet cutoff.
Using regulators at all was controversial, at least initially. But they work, and that trumps ideological purity. And yeah–they reflect the fact that our calculation breaks down below a certain scale, and something has to account for that. So it’s not quite as weird as it looks initially… but it’s still a little weird.
Good times.
Actually, your thread was from 2014. If anyone still can’t get enough of 1+2+3+4+… = -1/12, there’s an even more recent thread from 2015 (that links back to those other two).
I’ve always thought of infinity as an adjective, and not a quantity.
So although the question “how big is infinity?” sounds linguistically ok, it’s as meaningless a question as “how big is circular?”.
Thank you! I think I see, from The Great Unwashed’s example, how this can be manipulated to give any result desired. It appears to be an example of subtracting infinity from infinity, which isn’t defined.
Some other weirdness:
By the above, 1+2+4+8+… = -1
We can make an analogy to numbers that we’re allowed to extend infinitely to the left. The sequence above, in binary, is thus (hand-waving):
…1111111111[sub]2[/sub]
What happens if we add 1? Well, the right-most digit becomes 0, and we carry a 1, which we add to the second-right-most digit, and so on. So after adding 1 we have:
…0000000000[sub]2[/sub]
And our sum was -1, which if we add 1 to gives 0. So it seems that we’ve correctly computed -1+1=0. Computer geeks will see something familiar here.
It works for other bases. 1+3+9+27+… = -1/2, and 21+23+2*9+… = -1. So, in ternary:
…2222222222[sub]3[/sub] = -1
Add 1 again, and we get …0000000000[sub]3[/sub] as before. How about 61+67+649+… = 6 * -1/6. Then:
…6666666666[sub]7[/sub] + 1 = 6-1/6 + 1
Yet again, 0=0.
No, that’s not true. If you set a few rules and follow them, there is generally only one possible sum that results. It is certainly the only sum that has the linearity property. And there are generally a few different ways one can do the sum, all of which give the same answer.
It is definitely not the same as subtracting infinity from infinity, which is undefined. The results here are useful and behave consistently. But you have to generalize the idea of a sum a little bit, from “this is all the numbers added together” to “I associate this number with this sequence. The number behaves almost identically to the original sequence and so I can sometimes treat them as synonyms.”
If you like, just forget the word sum completely. Instead, call it a schlum. I define a schlum of a sequence a like this:
If a sequence is convergent, then the schlum is equal to the sum of the sequence.
If a sequence is divergent, then a schlum may or may not exist. If it exists, then it is the unique number that has the same properties as the sequence, such as aA = aschlum(A), and schlum(A)+schlum(B) = schlum(A+B).
There–you no longer have to get over the hurdle of thinking of this as a bunch of numbers added together. It’s not; it’s just a number that behaves kinda-sorta like a sum would have.
All these replies and no Buzz Lightyear quote? Hmmph! 
We were all thinking it. 
We’re having enough trouble getting to infinity, let alone beyond it. 
ETA: Infinity plus one - Wikipedia :eek:
I worked out a fun one!
A = 1 + 2 + 3 + 4 + 5 …
Let’s add 1 to each term: This involves adding an infinite number of “1s” to A.
B = (1+1) + (2+1) + (3+1) + (4+1) + (5+1) …
B = 2 + 3 + 4 + 5 + 6 …
B is obviously exactly the same as A, less the first term, “1.”
So…by adding in infinite number of 1’s to A…I have decreased its value by 1.
Clearly this kind of manipulation is not good math!
Nope, because you can still match the set of integers one-for-one with the set of integers omitting 1:
A1,B2 A2,B3 A3,B4 and so forth.
Again, you should probably stop thinking of these as literal sums. We already know that the sum of 1+2+3+… increases without bound, so it’s not too surprising that adding in another divergent sequence (1+1+1+…) would have similar behavior.
You did highlight a problem, though. In fact it can be shown more simply:
A = 1+1+1+…
A = 1 + (1+1+1+…)
A = 1 + A
Contradiction! What this demonstrates, among other things, is that you aren’t allowed to just arbitrarily shift sequences left and right and add/subtract them. You may do so partially, under some conditions–specifically, the summing method must have a property called stability, which basically says that we’re allowed to peel off the first number and add it to the sum of the rest.
The steps above only make use of the stability property, but it leads to a contradiction–which tells us that no stable summation method exists that works for 1+1+1+… For example, the method I outlined in post #18 cannot work, since it gives a 0 in the denominator.
There are some more advanced methods which assign a sum of -1/2 to 1+1+1+…, but they are not stable. Hence, the manipulations you used would be against the rules and so not demonstrate a contradiction.
(Technically, stability is a property of the series and not the summation method. It’s just that some summation methods only work for stable series and fail for unstable ones)
It’s like saying, "Is going to Ghana a journey or destination. Brother it’s the same thing because what does the plane or train say when you’ve finished a journey.“You have reached your destination!” Sorry this is too funny.
I dunno, perhaps you’ve proved that 1 + 1 + + 1… = -1
Like you. I have similar misgivings about the hi-falutin’ rearrangement of infinite series BUT every case I’ve tried seems to be “internally” consistent.
For instance I tried adding
1 - 2 + 3 - 4... = 1/4
to
1 - 1 + 1 - 1...= 1/2 (i.e. one term "late")
Giving
1 - 1 + 2 - 3 + 4... = 3/4
Aha! I thought, the bit starting - 1 + 2 - 3 + 4… is just -(1 - 2 + 3 - 4…) so is equal to -1/4
And then DAMNED consistency, 1 - 1/4 does indeed equal 3/4.
I await your counter-example with impatience.
(I’ve found one myself, add the 1 + 1 + 1… as you did but TWO terms late, you get 1+2+4+5+6… obviously A less 3. So there you have it, 1+1+1… = -1 AND -3)