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Sum of all numbers in -1/12

I came across this video that has now put my mathematical mind in a loop trying to think of why that 1+2+3+4 all the way to infinity = -1/12

Anyone seen this before or has an explanation as to why?

https://www.youtube.com/watch?v=w-I6XTVZXww

Reply 1

Phichi

Original post by Tiri

I came across this video that has now put my mathematical mind in a loop trying to think of why that 1+2+3+4 all the way to infinity = -1/12 Anyone seen this before or has an explanation as to why?https://www.youtube.com/watch?v=w-I6XTVZXww

AFAIK all proofs hinge on treating the infinite sum of Grandi's series as

\frac{1}{2}

, despite the fact it's divergent. Read more about it here:

http://en.m.wikipedia.org/wiki/Grandi%27s_series

Reply 2

Noble.

S_n = \underbrace{1 -1 + 1 -1 + ... \pm 1}_{\text{n times}}

does not converges to

\frac{1}{2}

, there doesn't exist a

N \in \mathbb{N}

such that for all

n > N

|S_n - \frac{1}{2}| < \epsilon

for

\epsilon \leq \frac{1}{2}

so it doesn't converge by the standard definition of convergence.

Reply 3

TeeEm

Original post by Tiri

1+2+4+8+16+32+ ....= a/(1-r) = 1/(1-2) = -1 which is a loads of rubbish in analogy as the sum to infinity is used incorrectly outside its radius of convergence ...

Reply 4

rayquaza17

Original post by Noble.

S_n = \underbrace{1 -1 + 1 -1 + ... \pm 1}_{\text{n times}}

does not converges to

\frac{1}{2}

, there doesn't exist a

N \in \mathbb{N}

such that for all

n > N

|S_n - \frac{1}{2}| < \epsilon

for

\epsilon \leq \frac{1}{2}

so it doesn't converge by the standard definition of convergence.

Should it not be

\epsilon >0

Reply 5

Farhan.Hanif93

The Cesaro sum is not the same thing as the 'actual' infinite sum, which is clearly divergent. The video is therefore immediately nonsense when they fail to distinguish the two.

Reply 6

Noble.

Original post by rayquaza17

Should it not be

\epsilon >0

If you wanted to show convergence then you'd have to show the existence of an N for any

\epsilon >0

. If you have some

\epsilon

such that no such N exists, then you have shown it's not convergent to that limit. All I'm saying by

\epsilon \leq \dfrac{1}{2}

is that there are (uncountably) infinite

\epsilon

convergence fails for, so it can't possibly converge to 1/2

(edited 9 years ago)

Reply 7

omegaSQU4RED

To be honest, I think any argument that the sum is -1/12 is a load of rubbish. If you have a series of non-negative terms, then how on earth is it possible for that sum to be negative?!

Reply 8

Tiri

Original post by omegaSQU4RED

To be honest, I think any argument that the sum is -1/12 is a load of rubbish. If you have a series of non-negative terms, then how on earth is it possible for that sum to be negative?!

Logistically it makes no sense. I think the video is just a physicist who thinks he can 'out do' mathematics.

Original post by omegaSQU4RED

To be honest, I think any argument that the sum is -1/12 is a load of rubbish. If you have a series of non-negative terms, then how on earth is it possible for that sum to be negative?!

It's a load of rubbish for Cauchy summation. However, it can take any value we like, depending on how we choose to define "summation". In the video, they chose to define:

\displaystyle\sum^{\infty}_{n=1} a_n = \displaystyle\lim_{m\to \infty} \dfrac{1}{m}\displaystyle\sum_{n=1}^{m}(a_1+a_2+…+a_n)

Which is where the 1/2 for the alternating 1-series arises. This is all fine and good, and may have applications in string theory or whatever, but that choice of definition is arbitrary and is clearly not the same thing as summing the series in the conventional sense (which can't be done). Hence by making another choice, we could clearly make the series take another value.

It's very bad form for them to not be clear on which definition of sum they are using, and astoundingly claim that this is the same as summing it normally.

Reply 11

omegaSQU4RED

Original post by Tiri

Logistically it makes no sense. I think the video is just a physicist who thinks he can 'out do' mathematics.

Once again, indicating that physics wouldn't make any sense without mathematics. There are also people who think they can "outdo" mathematics by proving that 1 = 2, except their proof somewhere will involve some caveat such as division by zero (which any sane person knows is impossible).

The moral of the story: Do not confuse reasoning that sounds good with good sound reasoning.

Reply 12

davros

Original post by Tiri

I swear we had this about a week ago!

http://www.thestudentroom.co.uk/showthread.php?p=54520849&highlight=zeta

:smile:

Reply 13

dknt

Most proofs of this you see will be fudged to get to the right answer. They're only there to demonstrate it to people with little maths knowledge. The actual proof I believe is done via the analytic continuation of the Riemann zeta function for s = -1, which will give the required result. It's a way of assigning a sensible value to the series.

(edited 9 years ago)

Reply 14

Tiri

Original post by davros

I swear we had this about a week ago!

http://www.thestudentroom.co.uk/showthread.php?p=54520849&highlight=zeta

:smile:

Ah I didn't see this. Im just a novice mathematician (If that), who was just curious about this proof. I have learnt though from this debate, that mathematicians and physicist approach maths very differently.