What is the Significance of the -1/12 Sequence in Physics?

[Null_]

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?

 

[micromass]

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 .

 

[Null_]

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!

 

[gb7nash]

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.

 

[Mentallic]

If he thinks that, he needs to get his head checked.

Agreed

 

[diazona]

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
$$\sum_{n=1}^\infty \frac{1}{n^s}$$
for $s > 1$, and then create an analytic function $\zeta(s)$ that produces the same values, and look for the value of that function at $s = -1$.

 

[atyy]

http://math.ucr.edu/home/baez/numbers/

 

[dimension10]

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?

 

[disregardthat]

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.