OT: 1+2+3+4+5.... = not what you'd think.

David Rutten

david@mcneel.com

Replies to This Discussion

Permalink Reply by Hannes Löschke on January 9, 2014 at 2:37pm

Okay, I'm not a math crack, but there are some points that confuse me:

a) the result of an infinite sum is where it converges to. Since S is clearly divergent, whatever result you might get is invalid, since it doesn't apply to the given sum.

b) there is a clear difference between "equals to" and "converges to", that is ignored here right from the beginning.

c) the -1, 1 sequence never actually converges. It oscillates around 1/2. So ignoring the converging part you could state 1/2 was the result, but it's actually not. Same is true for Zeta(-1).

d) the 1-2+3... sequence actually diverges too, while it oscillates around 1/4

Since S is divergent it has no valid result. Any multiplication or subtraction is undefined. The final equation is like having both sides divided by x then solved for x with the result of 0.

Permalink Reply by David Rutten on January 9, 2014 at 3:00pm

There are better proofs for (c), the video (when watched on YouTube) links to those. As for the rest, I'm thoroughly confused too, though I can't fault the proof. I suppose it's just the answer if you make certain assumptions in your mathematical system, which you always have to do anyway.

David Rutten

david@mcneel.com

Permalink Reply by Mateusz Zwierzycki on January 9, 2014 at 4:03pm

I think that Hannes has a good point here... other sequences (+1-1..) are oscillating around some number, which doesn't mean they are = to some number in all cases, the equality sign here is just a matter of notation. What scares me most here is the roundness of -1/12... i would expect Pi as a result ;)

Permalink Reply by David Rutten on January 10, 2014 at 1:41am

Seriously, watch the proof for 1-1+1-1+1-1+1-1+... = 0.5, it's pretty convincing.

David Rutten

david@mcneel.com

Permalink Reply by Danny Boyes on January 10, 2014 at 2:28am

The bulb would blow... just like my mind :)

Permalink Reply by Mateusz Zwierzycki on January 10, 2014 at 6:50pm

I'm not saying the mathematician is wrong, I think that the math is wrong here ;)

After watching 1+1-1... - who said that you can use brackets whereever you want ?

This is so wrong :

1 + 1/2 + 1/4 + 1/8 .... 1/∞

Let's reverse the sequence and use common denomitator.

1/∞ + 2/∞ + 4/∞ + 8/∞ .... ∞/∞

So 1 = ∞/∞

Also take a look at this marvelous proof for 1/∞ = 0.

I think that we can safely assume 1/∞ = 0. Let's go back to our sequence :

1/∞ = 0

2/∞ = 2*1/∞ = 2*0 = 0

4/∞ = 4*1/∞ = 4*0 = 0

∞/∞ = ∞*1/∞ = ∞*0 = 0

therefore

1 = 0

Permalink Reply by David Rutten on January 11, 2014 at 2:42am

Ignore the bit about the brackets, that's indeed a weird tangent that doesn't make sense. It's the 1-S = S bit that's proof for S = 0.5

David Rutten

david@mcneel.com

Permalink Reply by Hannes Löschke on January 11, 2014 at 2:29am

So there you have it. Even in math, a result is not always a solution. Infinite sums cannot be solved. All you can do is estimate the result of a non divergent series by finding its limit towards infinity. While you can find an answer to

The S1 of the series 1-1+1-1+1... doesn't actually converge, you cannot find an answer. You need to transform it the sum of partial averages to find a solution for this. But as we do this, the series is no longer 1-1+1-1+1... the transformation obviously erases valid solutions. Its like dividing an equation by x. You will erase 0 as a possible solution for x.

Since the original alternation pattern isn't there anymore, we can no longer add the parts of the original sum. All we can do is add the partial sum of S and the partial average while the partial sum will diverge and the partial average converges to a constant, the partial sum of both will always diverge.

-1/12 is not the solution. It's an artifact of the math involved.

The answer to the light bulb is actually pretty simple: its on for more than two thirds of the time until 2 minutes and after that its undefined because we just left the scope of its on/off function.

Permalink Reply by Thomas on January 13, 2014 at 4:29am

it's just string theory ;)

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

Permalink Reply by Vicente Soler on January 13, 2014 at 7:06am

In the video they also only mentioned string theory sort of implying that the math is not wrong because you can find it in models that predict the natural world, but afaik string theory is a mathematical, not scientific theory, it currently can't be tested, so I found it a bit misleading.

But apparently the Riemann zeta function is used in quantum mechanics in other models that have plenty of experimental evidence that they work.

Permalink Reply by David Rutten on January 13, 2014 at 7:15am

Yeah, I concur that ST ought not be counted amongst the physical sciences just yet, I'd like to see some falsifiable predictions first. Lee Smolin's and Jim Baggott's positions on this subject are correct in my opinion.

David Rutten

david@mcneel.com

Permalink Reply by Hannes Löschke on January 14, 2014 at 2:30am

Refreshing to see that some physicists actually remember that just math provides tools to model physics and you still need prove the model or it's predictions to actually have a meaning in our real world.