Can Summing Infinite Series to -1 Be Valid?

[EternityMech]

http://www.youtube.com/watch?v=kIq5CZlg8Rg&feature=player_embedded

claims that if you had all the sums upto infinty you get -1

i think he's an idiot but it seems he's a theoretical phycisist

 

[Stephen Tashi]

He's not on to anything new in mathematics since it is already known that you can't draw valid conclusions about divergent series by regrouping their terms. Is he making a useful analogy to something done by quantum physicists (who are much less fussy about divergence than mathematicians)? I don't know. Perhaps a forum member who is a quantum physicist will tell us.

 

[Mark44]

The title should be "Is this guy on something?"

Here is what he has:
1 + 2 + 4 + 8 + ... = 1*(1 + 2 + 4 + 8 + ...)
= (2 - 1)(1 + 2 + 4 + 8 + ...)
= 2 + 4 + 8 + 16 + ... - 1 - 2 - 4 - 8 - ...
This is the step where it breaks down. The line above is essentially $\infty - \infty$, which is what is called an indeterminate form, along with 0/0 and several others.

If he were dealing with convergent series (he isn't), normal arithmetic would be applicable. Both series are divergent, though, so doing arithmetic with them leads to an erroneous result.

 

[phinds]

It is possible to be a theoretical physicist and still be an idiot but in this particular case I don't think he is ... he's just playing a mind-game on folks who don't understand the arithmetic of infinity. Basically, he's saying 2 times infinity is different than infinity but he hides it by not using the infinity symbol but instead using the series and pretending that he can treat the series differently that he would treat the infinity symbol.

You can prove anything if you play invalid games with zero and infinity.

EDIT: I see Mark44 and I were typing at the same time and his statement "Both series are divergent, though, so doing arithmetic with them leads to an erroneous result." is a more elegant explanation than mine (but amounts to the same thing)

 

[Mute]

Perhaps watching the follow-up video (linked to at the end of the movie, but I'll link here as well) will answer some questions: video link

If you're left with more questions than answers after that (which you likely will be), the 'technique' which physicists use is called "regularization" or "zeta regularization" in some specific instances.

The basic idea is that sometimes when you run into divergent sums in your calculations (in physical problems), they're really not supposed to be divergent sums - they should be something else that's finite, but due to approximations or the theory being incomplete you get this divergent beast. The regularization is a trick to replace the divergent sum with something finite, which is what the sum is "supposed to be".

 

[SteveL27]

I didn't watch the vid but is this referring to Ramanujan summation?

http://en.wikipedia.org/wiki/Ramanujan_summation

For example, 1 + 2 + 3 + ... = -1/12. Apparently this means something to some people but unfortunately it's beyond my level.

 

[tahayassen]

http://en.wikipedia.org/wiki/1_+_2_+_4_+_8_+_...

Everyone in this thread says he's wrong, but the Wikipedia article says he's right. Any insight?

Also, check out this book: http://www.archive.org/stream/divergentseries033523mbp#page/n35/mode/2up

 

[D H]

He's not on to anything new in mathematics since it is already known that you can't draw valid conclusions about divergent series by regrouping their terms.

So don't do that then.

If he were dealing with convergent series (he isn't), normal arithmetic would be applicable. Both series are divergent, though, so doing arithmetic with them leads to an erroneous result.

Sure you can do arithmetic with divergent series. Euler lead the way. Hardy wrote the book. G.H.Hardy, Divergent Series