1 + 2 + 3 +... = -1/12

[Leonhard]

In another thread the prospect of infinite sums came up and I wanted to introduce one of the oddest results I've ever seen in math. Its a situation where clearly the partial sums tend towards infinite, and where its possible to develops means of summing up the infinite numbers in such a way that it returns a specific result.

Let's dive right into it and give the result.



This ridiculously counterintuitive result stems from a field called Ramanujan summanation, which is one in a group of summation methods for infinite series that are divergent. Strangely enough some of these methods can be used to sum up the infinite series of integers, and they always end up given the result above.

I'll show you how this result can be derived from something called Abel summation.

Now if we take another series, namely



Then naturally we'd try to solve this by finding its partial sum (summing up all but the leading term) and seeing what this converges to



Obviously the partial sum is divergent as we add more and more terms. So we'll need a different way of summing. The mathematician Abel proposed (though Euler found it first), the following means of handling this type of summation.



This can't sum the first series mentioned, but it is able to sum the second one, giving us...



From this we can determine what series mentioned in the beginning will be, if it has any value at all. This is done simple be substracting the partial sum of one series from the other.



Of course this result can also be found directly, but that requires a stronger method of summation than Abel summation, such as Ramanujan summation. However unsurprisingly it yields the exact same answer. And its not merely a theoretical answer, as sums over all the natural integers occur often in quantum field theory, and in the derivation of the Casimir effect one has to use such a sum.

Therefore this result, along with various strong derivations are included in Advanced Quantum Mechanics course work.

What do you guys think?

[37818]

I'm of the opinion that it is a special case. Modifying the sum, how it is derived. That being said:

[Leonhard]

Yes you can't do it with all series, there are series such that all the summation methods that we know will fail to make a definite result of them. And really it shouldn't be thought of as a proper summation. Results such as -1/12 should be thought of as representing a property of the partial sums, rather than really representing the sum of all integers.

[37818]

Yes you can't do it with all series, there are series such that all the summation methods that we know will fail to make a definite result of them. And really it shouldn't be thought of as a proper summation. Results such as -1/12 should be thought of as representing a property of the partial sums, rather than really representing the sum of all integers.

Yet it is still the limit result of an infinite series taken it that special why. And it seems to be a relevant part in String Theory.

[Leonhard]

Yet it is still the limit result of an infinite series taken it that special why. And it seems to be a relevant part in String Theory.

So far there's no realistic interpretation of why applying what's called 'regulators' works out in quantum field theory (not just in String Theory). Its a mystery that has yet to be completely solved, but there's work in progress on it.

Personally I'm not fond of String Theory, but for other reasons I won't go into.

[robrecht]

See the exchange between lao tzu and me that starts here: http://www.theologyweb.com/campus/sh...ll=1#post45840

I think this result is caused by the artificial introduction of randomness into the 'idea' of infinity.

[37818]

See the exchange between lao tzu and me that starts here: http://www.theologyweb.com/campus/sh...ll=1#post45840

I think this result is caused by the artificial introduction of randomness into the 'idea' of infinity.

Where and how is the randomness introduced? A fixed pattern is hardly random.

[Darth Executor]

Does this mean that if you travel with infinite speed you start moving back in time?

[robrecht]

Where and how is the randomness introduced? A fixed pattern is hardly random.

Watch the video, starting around 2 minutes. The sum of S1, depending on where you stop, will either be one or zero. Since they cannot determine what number will be stopped at, they take the average, which is 1/2. That's randomness. In 'reality', if you are counting an infinite number of numbers, you do not stop anywhere, but keep going on forever.

[Boxing Pythagoras]

Watch the video, starting around 2 minutes. The sum of S1, depending on where you stop, will either be one or zero. Since they cannot determine what number will be stopped at, they take the average, which is 1/2. That's randomness. In 'reality', if you are counting an infinite number of numbers, you do not stop anywhere, but keep going on forever.

That's not randomness. That's taking an average over a field of values. Nothing random about it.

[Leonhard]

Watch the video, starting around 2 minutes. The sum of S1, depending on where you stop, will either be one or zero. Since they cannot determine what number will be stopped at, they take the average, which is 1/2. That's randomness. In 'reality', if you are counting an infinite number of numbers, you do not stop anywhere, but keep going on forever.

Maybe 'ad hoc' is a better word to describe what you mean than randomness?

[robrecht]

That's not randomness. That's taking an average over a field of values. Nothing random about it.

An average based on probability rather an arbitrary choice, which in 'reality' should not be made in a sum of a 'truly infinite' number of numbers.

[robrecht]

Maybe 'ad hoc' is a better word to describe what you mean than randomness?

Perhaps. I'm not sure of what technical definitions of 'randomness' might or might not be used here. What seems random or arbitrary is that the average is based on one or another number (one or zero) depending on where in the series one might 'stop', with a 50% chance of either sum being the result. This seems artificially introduced into the idea of infinity, according to a mathematical attempt to come to terms with infinity.

[Leonhard]

Perhaps. I'm not sure of what technical definitions of 'randomness' might or might not be used here. What seems random or arbitrary is that the average is based on one or another number (one or zero) depending on where in the series one might 'stop', with a 50% chance of either sum being the result. This seems artificially introduced into the idea of infinity, according to a mathematical attempt to come to terms with infinity.

Oh it has nothing to with infinity as such! Its just that when you have an infinite series, there's more than one way of summing it up. You're quite correct that the partial sum diverges (jumping between 1 and 0), and so you can't sum up the infinite series by looking at the limit of the partial sum. Which is what you'd typically do.

Instead what you could do is you introduce a regulator, which is what is done in Abel summation. Its a small factor multiplied to each member of the series, and then we look at the limit as this factor goes towards one. This way of summing up the series produces the answer 1/2.

[Carrikature]

Now if we take another series, namely



Then naturally we'd try to solve this by finding its partial sum (summing up all but the leading term) and seeing what this converges to

You're not actually summing up all but the leading term the way you've shown it. The summation would go to negative infinity given that each value subtracted is higher than the number added directly before it: (1-2)+(3-4)+(4-5)... With each pair, you're shifting the summation further negative by 1.

Obviously the partial sum is divergent as we add more and more terms. So we'll need a different way of summing. The mathematician Abel proposed (though Euler found it first), the following means of handling this type of summation.



This can't sum the first series mentioned, but it is able to sum the second one, giving us...

I'd hardly say the partial sum is 'obviously' divergent, but I'm not really worried about that part here. If a proposed solution for summing integers results in a fraction, I'd say that solution isn't any solution at all. Abel's summation might work where the integers are positive, but it seems to not apply when those integers are negative. You've also made up a completely new series and taken to calling it part of the original series.

From this we can determine what series mentioned in the beginning will be, if it has any value at all. This is done simple be substracting the partial sum of one series from the other.



Of course this result can also be found directly, but that requires a stronger method of summation than Abel summation, such as Ramanujan summation. However unsurprisingly it yields the exact same answer. And its not merely a theoretical answer, as sums over all the natural integers occur often in quantum field theory, and in the derivation of the Casimir effect one has to use such a sum.

Therefore this result, along with various strong derivations are included in Advanced Quantum Mechanics course work.

What do you guys think?

This makes no sense to me. You created a second series, used a summation method that seems to give an obviously false result, and then proceeded to subtract the second series from the first in an attempt to solve its summation.