Wait what? (1+2+3+4+...

[BAndrew]

I am talking about these videos:
https://www.youtube.com/watch?v=w-I6XTVZXww
http://www.youtube.com/watch?v=E-d9mgo8F...e=youtu.be

It suggests that 1+2+3+4+5+... = -1/12.

Spoiler below!

OK, this is ridiculous (I don't care if physicists use it, I only care about it mathematically) and I doubt it's mathematically correct for the following reasons. If someone knows about the subject please inform me, because I think this is some bs. So here we go:

Objection 1:
Proof of first video:
S = 1+2+3+4+5+... is not a valid statement because he is assuming that the sum has a real value and he is doing operations that are true only in R (real number set).
If the sum was infinite then when he says 2S-S it would be ∞-∞ which is undefined and not S.

Objection 2:
Proof of second video:

• What he writes at some point is true for |x|<1 AND NOT x<1 and therefore he can't substitute x= -1 as he later does.
  
• Wikipedia clearly states here that:
  Re(s) > 1 and again he can't substitute x = -1 because Re(-1) = -1<1
  

It gets even worse...

Objection 3:
Another youtuber clearly using a similar method arrives to the conclusion that 1+2+3+4+... = -1




So obviously -1 =/= -1/12

Therefore something is really wrong here.

Objection 4:
I can use the same method to prove that 0 = -1

Let S = 1+2+3+4+...
then 2S = 2+4+6+8+10+...

S = 2S-S = 2+4+6+8+... -(1+2+3+4+...) = 2+3+4+5+6+... = S-1


Hence S = S-1 =>
0 = -1


Good job, Einstein!
Obviously this is not true. If physicists use this then I am really dissapointed and I have lost all the respect I had for them.

Am I right or what?

NOTE: I have changed opinions since I opened this thread. Apparently I rushed on making conclusions.

[Bridge]

I would say the same thing about 0.999… = 1, which you seem to think is sound math. Garbage in, garbage out.

[BAndrew]

I would say the same thing about 0.999… = 1, which you seem to think is sound math. Garbage in, garbage out.

What has this anything to do with 0.9999... = 1?
0,999...= 1 has a valid proof that everyone accepts and creates no problem.

But this is a total mess. I even proved that 0 = -1 with the same logic. That means there is something wrong with the proof(unless someone believes that 0 = -1).

[Ghieri]

Regarding: .99... = 1. It is largely considered true by mathematicians and the proof is solid.

Regarding the current problem: The very second they mentioned string theory I closed the video. That stuff hurts my brain something fierce. I have a feeling even after they explain it I'll still be going "Huuh?!"

[BAndrew]

Regarding: .99... = 1. It is largely considered true by mathematicians and the proof is solid.

Regarding the current problem: The very second they mentioned string theory I closed the video. That stuff hurts my brain something fierce. I have a feeling even after they explain it I'll still be going "Huuh?!"

They better have a very good explanation for it, but I don't think they do. Look at all the stuff I exposed.

[PutraenusAlivius]

http://boards.straightdope.com/sdmb/show...p?t=420045

[BAndrew]

http://boards.straightdope.com/sdmb/show...p?t=420045

It doesn't really help.

It mentions analytic continuation which explains why he can put x = -1. But even if that is true then why I get results like 0 = -1? and -1 = -1/12?

[Froge]

Shifting the second S2 to the right so that adding them together recreates S1 is not valid. Let's say S2 grows very large (i.e. it approaches infinity). The fact that the second S2 has been "shifted" to the right implies that it is always one term ahead of the first S2, and this term cannot be neglected even as the sums approach infinity. This can be best illustrated by pretending there is a stop point. The magnitude of the last term of the second S2 is tremendous as S2 approaches infinity so it makes a big difference.

To show an example, let's pretend S2 stops at 10^99:

S2a = 1 - 2 + 3 - 4 + ... + (10^99 - 1) - 10^99
+ S2b = 0 + 1 - 2 + 3 - .... - (10^99 - 2) + (10^99 - 1) - 10^99
-------------------------------------------------------------------------------
1 - 1 + 1 - 1 + ... + 1 - 1 - 10^99 = -10^99

Notice the -10^99 that wasn't added? I know the "well it should equal S1 when it reaches infinity" but that argument doesn't apply to a series of natural numbers, because S2b will always contain one more term than S2a. Even if the number of terms in the first S2 approach infinity, the number of terms in the second S2 approach a sort of "infinity plus one" (i.e. still one more term than the first S2), which of course makes a giant difference in the sum.

I also think the very first sum S1 isn't correct. 1 - 1 + 1 - 1 + 1 - 1 + ... does not equal 1/2. Not only that, but I don't think the guy doing the proof in the video knows his terminology. He says 1/2 is the "natural number" that we attach to the sum S1, but 1/2 is definitely not a natural number. Of course this is more of an ad hominem rebuttal, but I think it decreases his credibility.

[Bridge]

0,999...= 1 has a valid proof

So, apparently, does this. It doesn't matter whether it's "accepted" if it is clearly erroneous. Anybody can tell you that 1 + 2 + 3 ... = -1/12 violates a fundamental mathematical principle: two natural numbers added together will always make a natural number. Similarly, 0.999... = 1 violates another principle which states that if you have a number with nonzero decimals it is not a natural number. These two are so obvious they scarcely need to be said. Why then do you accept illogical data that is the result of making calculations in systems that do not support said calculations?

EDIT: 0.999... doesn't really have anything to do with the thing you posted, I just hoped I could take the chance while you were in a skeptical mindset to make you rethink it. I still persist that the proofs offered to support it are facile.

[Froge]

Why then do you accept illogical data that is the result of making calculations in systems that do not support said calculations?

0.999... is not a number with nonzero decimals, despite how it looks when it is written. A basic principle of mathematics is that there is some distance, no matter how infinitesimal, between any two different numbers. For example, there are an infinite quantity of numbers between 0.99 and 1 (eg. 0.991, 0.9999992, etc.) However, there is absolutely no distance between 0.999... and 1 because the 9s go on to infinity.

Fundamentally, you have to realize that the 9s simply do not stop. You cannot assume "oh, even after you go on for a really long time eventually the 9s end" because that is not the definition of infinity. Infinity is a concept defined by that "if there is a number, infinity is larger." So even the largest numbers ever conceived (example: graham's number) are infinitely small compared to infinity. That means no matter how you try, you absolutely cannot find a value between 0.999... and 1.

Think of 0.999... as just a different way of writing 1. It is an application of the concept of infinity.