Exploring the -1/12 Sequence: A Physicist's Analysis

In summary, a physicist mentioned a sequence from n=1 to infinity with an apparent answer of -1/12. However, this answer is not the true sum of the series, which actually diverges or has an infinite sum. Instead, the Ramanujan sum is used to represent this series, which has useful properties but is not the conventional sum. This concept can also be related to zeta function regularization and has sparked discussions about the true value of 1+2+3+... in different contexts.
  • #1
Null_
231
0
I attended a talk where a physicist mentioned this sequence from n=1 to infinity and apparently the answer is -1/12? Could someone explain please?
 
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  • #2
Well, of course the answer isn't really -1/12, rather, the answer is that the series diverges. Or maybe that the sum is infinite.

However, to some divergent series, one still can associate a number (called: the Ramanujan sum). This Ramanujan sum is not the sum of the series in the conventional sense, but rather a substitute for the conventional sum which still has a lot of useful properties.

So, in a way, it is true that 1+2+3+...=-1/12. But one should always specify that we're working with Ramanujan sums instead of conventional sums. That's all I know from this, more information on http://en.wikipedia.org/wiki/Ramanujan_summation .
 
  • #3
Thanks for the explanation and the link. I'm in Calc II now and we're currently learning series. Everything seems pretty obvious that we've done, so I was surprised to hear his statement.

I'll be browsing wikipedia tonight to learn more about sequences and series!
 
  • #4
Null_ said:
I attended a talk where a physicist mentioned this sequence from n=1 to infinity and apparently the answer is -1/12? Could someone explain please?

If he thinks that, he needs to get his head checked. He was probably joking.

Nevermind, once again I learn something new. I've never seen a Ramanujan sum before.
 
  • #5
gb7nash said:
If he thinks that, he needs to get his head checked.

Agreed :biggrin:
 
  • #6
It might be worth mentioning that the same answer comes from zeta function regularization, which seems like it might be a little easier to understand. In that technique you compute
[tex]\sum_{n=1}^\infty \frac{1}{n^s}[/tex]
for [itex]s > 1[/itex], and then create an analytic function [itex]\zeta(s)[/itex] that produces the same values, and look for the value of that function at [itex]s = -1[/itex].
 
  • #8
micromass said:
Well, of course the answer isn't really -1/12, rather, the answer is that the series diverges. Or maybe that the sum is infinite.

However, to some divergent series, one still can associate a number (called: the Ramanujan sum). This Ramanujan sum is not the sum of the series in the conventional sense, but rather a substitute for the conventional sum which still has a lot of useful properties.

So, in a way, it is true that 1+2+3+...=-1/12. But one should always specify that we're working with Ramanujan sums instead of conventional sums. That's all I know from this, more information on http://en.wikipedia.org/wiki/Ramanujan_summation .

But isn't itsupposed to be a form of zero proof?
 
  • #9
While 1+2+3+... could be said to equal anything in the right context, -1/12 is interesting because the zeta-function is the unique analytical extension of the sum in diazona's post.
 

1. What is the -1/12 sequence and why is it significant in physics?

The -1/12 sequence, also known as the Ramanujan summation, is a mathematical series that appears to sum to the value of -1/12 when using a particular method of summation. It is significant in physics because it has been used in various mathematical models to describe physical phenomena, such as in string theory and quantum field theory.

2. Is the -1/12 sequence a real number or just an abstract concept?

The -1/12 sequence is a real number, as it can be expressed as a decimal approximation (-0.083333...). However, it is also an abstract concept in the sense that it is a result of a mathematical operation and does not have a direct physical interpretation.

3. How is the -1/12 sequence derived and what is the significance of its derivation?

The -1/12 sequence can be derived through various mathematical methods, including zeta function regularization and Ramanujan summation. Its significance lies in the fact that it has been used in physical theories and calculations, and its derivation can lead to a deeper understanding of these theories.

4. Can the -1/12 sequence be used to solve real-world problems?

While the -1/12 sequence has been used in physical theories, it is not typically used to solve real-world problems. Its use is primarily in theoretical and mathematical contexts, rather than practical applications.

5. Is the -1/12 sequence a controversial concept in the scientific community?

The -1/12 sequence has been a topic of debate and controversy in the scientific community, with some arguing against its validity and others advocating for its use in certain theories and calculations. However, its use has been supported by mathematical proofs and has been used successfully in various contexts, leading to ongoing discussions and research in the scientific community.

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