# 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

Math is amazing and such a big *bleep* sometimes!

I remember this one time my teacher in class explained to us something which basically said that the number 1 is actually not equal to one but to 0.999... Since 1/3 is 0.333... And 2/3 is 0.666... Then 3/3 (which is another way of writing 1) should be 0.999... and not exactly 1.

These kinds of things always make your mind explode! You think you know something.. but you don't.

Oh, and numberphile , definitely one of the best and more interesting channels on YouTube!

I remember this one time my teacher in class explained to us something which basically said that the number 1 is actually not equal to one but to 0.999... Since 1/3 is 0.333... And 2/3 is 0.666... Then 3/3 (which is another way of writing 1) should be 0.999... and not exactly 1.

These kinds of things always make your mind explode! You think you know something.. but you don't.

Oh, and numberphile , definitely one of the best and more interesting channels on YouTube!

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

The .9999 = 1 thing at least makes intuitive sense, this is just a complete mindf***.

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

Can you really consider 1-1+1-1..... = 1/2??

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 +......

= 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)

= 1 - S

Therefore, 2S = 1 and S = 1/2

= 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)

= 1 - S

Therefore, 2S = 1 and S = 1/2

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

The one thing I would say is that his initial logic is flawed.

1-1+1-1+... =/= 1/2. It equals 1 OR it equals 0. The reality is that since infinity goes on forever, we can never know whether it equals 1 or 0, but just saying that it equals 1/2 on average since we'll never know the answer doesn't make it actually equal 1/2. 1/2 is just an assumption. From this point, it all falls apart.

I'm not saying that his message is flawed though, at least in terms of my understanding of it. For me, what this proves is that any attempt by humans to comprehend the extent of infinity is a lost cause. We were never meant to understand it, which is a mind-feck in and of itself of course.

If you are subtracting what S is equal to from 1, then the left side needs to be subtracted from 1 as well. You did this on the right side without doing it on the left side. The formula should read:

1-S = 1-(1 - 1 + 1 - 1 + 1 - 1 + .....)

Therefore once we remove the 1, it becomes:

-S = -(1 - 1 + 1 - 1 + 1 - 1 + .....)

Which again, makes S equal 1 or 0. Not 1/2

1-1+1-1+... =/= 1/2. It equals 1 OR it equals 0. The reality is that since infinity goes on forever, we can never know whether it equals 1 or 0, but just saying that it equals 1/2 on average since we'll never know the answer doesn't make it actually equal 1/2. 1/2 is just an assumption. From this point, it all falls apart.

I'm not saying that his message is flawed though, at least in terms of my understanding of it. For me, what this proves is that any attempt by humans to comprehend the extent of infinity is a lost cause. We were never meant to understand it, which is a mind-feck in and of itself of course.

1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 +...... =/= 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)@Dutti wrote:S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 +......

= 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)

= 1 - S

Therefore, 2S = 1 and S = 1/2

If you are subtracting what S is equal to from 1, then the left side needs to be subtracted from 1 as well. You did this on the right side without doing it on the left side. The formula should read:

1-S = 1-(1 - 1 + 1 - 1 + 1 - 1 + .....)

Therefore once we remove the 1, it becomes:

-S = -(1 - 1 + 1 - 1 + 1 - 1 + .....)

Which again, makes S equal 1 or 0. Not 1/2

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

1 = 1. The only problem is that it's impossible to perfectly divide 1 by 3, so if you divide 1 by 3, then the 3 pieces no longer mathematically equal 1 when added together if you're using decimals, but that doesn't change the fact that 1 = 1.@Il Diavolo wrote:Math is amazing and such a big *bleep* sometimes!

I remember this one time my teacher in class explained to us something which basically said that the number 1 is actually not equal to one but to 0.999... Since 1/3 is 0.333... And 2/3 is 0.666... Then 3/3 (which is another way of writing 1) should be 0.999... and not exactly 1.

These kinds of things always make your mind explode! You think you know something.. but you don't.

Oh, and numberphile , definitely one of the best and more interesting channels on YouTube!

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

@Eman wrote:1 = 1. The only problem is that it's impossible to perfectly divide 1 by 3, so if you divide 1 by 3, then the 3 pieces no longer mathematically equal 1 when added together if you're using decimals, but that doesn't change the fact that 1 = 1.

That makes sense. But according to the way we define the division operation, the exact value of 1/3 is 0.333... And as a result of that 3/3 becomes 0.999... And not 1. All this really does is point towards anomaly in the fundamental definition of the division operation (which is that it is impossible to perfectly divide 1 by 3, or what you said). So you're right about this.

@Eman wrote:1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 +...... =/= 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)

If you are subtracting what S is equal to from 1, then the left side needs to be subtracted from 1 as well. You did this on the right side without doing it on the left side. The formula should read:

1-S = 1-(1 - 1 + 1 - 1 + 1 - 1 + .....)

Therefore once we remove the 1, it becomes:

-S = -(1 - 1 + 1 - 1 + 1 - 1 + .....)

Which again, makes S equal 1 or 0. Not 1/2

In this case, IMO S = 0.5 makes sense. Because when you bring S into the right side, you are not adding or subtracting anything from the equation. You are

**substituting**S for its numerical value. You only need to perform and operation on both sides of the equation if you are adding something to the equation that was not previously present, but in this case you are not.

1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)

Evaluating the parentheses;

1 - S = 1 - 1 + 1 - 1 + 1 - 1 + -1 +.....

Now since the right side = S, we substitute it for S (we did not add anything extra to the equation so we do not need to do this on both sides.)

--> 1 - S = S

--> S = 1/2

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

I know what you're saying, but the problem in your solution lies in the parts highlighted in red, and specifically the part in blue on the right side. In the equation I responded to, the '1 -' that had been added to the right side was not included on the left side. However, in your case where you correctly included it on both sides, in blue, it screws up your rationale that S = 1/2. To walk through step-by-step, if you start with the equation:@Il Diavolo wrote:@Eman wrote:1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 +...... =/= 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)

If you are subtracting what S is equal to from 1, then the left side needs to be subtracted from 1 as well. You did this on the right side without doing it on the left side. The formula should read:

1-S = 1-(1 - 1 + 1 - 1 + 1 - 1 + .....)

Therefore once we remove the 1, it becomes:

-S = -(1 - 1 + 1 - 1 + 1 - 1 + .....)

Which again, makes S equal 1 or 0. Not 1/2

In this case, IMO S = 0.5 makes sense. Because when you bring S into the right side, you are not adding or subtracting anything from the equation. You aresubstitutingS for its numerical value. You only need to perform and operation on both sides of the equation if you are adding something to the equation that was not previously present, but in this case you are not.

1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)

Evaluating the parentheses;

1 - S = 1 - 1 + 1 - 1 + 1 - 1 + -1 +.....

Now since the right side = S, we substitute it for S (we did not add anything extra to the equation so we do not need to do this on both sides.)

--> 1 - S = S

--> S = 1/2

"S = 1 - 1 + 1 - 1 + 1 - 1 +..."

And subtract each side from 1:

1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 +.....)

And make the amount on the right which S is equal to, S:

1 - S = 1 - S

You now get S = S, or 1 = 1.

1 - S = 1 - 1 - 1 + 1 - 1 + 1 - 1 +...

, will not give you the solution that 1 - S = S. It will give you 1 - S = 1 - S.

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

Anyway, I'm sure anyone reading this debate will just get super confused, but the moral of the story is that 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +... only equals -1/12 if an assumption is made that by its very nature cannot possibly be correct in the circumstances and is only made because we couldn't possibly determine what the answer is otherwise.

I would be interested in seeing what the 'more complex' approach mentioned in the video is to getting to this solution though.

I would be interested in seeing what the 'more complex' approach mentioned in the video is to getting to this solution though.

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

I think it gives both solutions, but since we are trying to isolate for S, we look at the one that lets us do that. But ultimately, infinity is a concept and not something that we can quantify or even imagine. So, I don't know. This is pretty confusing.@Eman wrote:I know what you're saying, but the problem in your solution lies in the parts highlighted in red, and specifically the part in blue on the right side. In the equation I responded to, the '1 -' that had been added to the right side was not included on the left side. However, in your case where you correctly included it on both sides, in blue, it screws up your rationale that S = 1/2. To walk through step-by-step, if you start with the equation:

"S = 1 - 1 + 1 - 1 + 1 - 1 +..."

And subtract each side from 1:

1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 +.....)

And make the amount on the right which S is equal to, S:

1 - S = 1 - S

You now get S = S, or 1 = 1.

1 - S = 1 - 1 - 1 + 1 - 1 + 1 - 1 +...

, will not give you the solution that 1 - S = S. It will give you 1 - S = 1 - S.

Here's the other (more complicated) video he was talking about:

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

The thing is that you can't isolate S on one side though without doing it on the other. Anyway, yeah basically that video just addresses the concern I had with the video itself. Basically what he says is that using a mathematical approach that would work for a set of numbers other than something like a bunch of 1s, you can come to the conclusion that the best answer is 1/2, but admittedly he says that it's not a real limit (i.e. not the real answer).

In complex logical situations that cannot be solved such as you might come across in physics, this is probably the best approach, but it's admittedly not precise, so expanding the logical equations to come up with insane answers doesn't really prove anything other than that math isn't always precise. The imprecise approach caused the answer to be -1/12; for me, what this does more than anything is discredit the approach.

In complex logical situations that cannot be solved such as you might come across in physics, this is probably the best approach, but it's admittedly not precise, so expanding the logical equations to come up with insane answers doesn't really prove anything other than that math isn't always precise. The imprecise approach caused the answer to be -1/12; for me, what this does more than anything is discredit the approach.

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

@Eman wrote:The one thing I would say is that his initial logic is flawed.

1-1+1-1+... =/= 1/2. It equals 1 OR it equals 0. The reality is that since infinity goes on forever, we can never know whether it equals 1 or 0, but just saying that it equals 1/2 on average since we'll never know the answer doesn't make it actually equal 1/2. 1/2 is just an assumption. From this point, it all falls apart.

I'm not saying that his message is flawed though, at least in terms of my understanding of it. For me, what this proves is that any attempt by humans to comprehend the extent of infinity is a lost cause. We were never meant to understand it, which is a mind-feck in and of itself of course.1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 +...... =/= 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)@Dutti wrote:S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 +......

= 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)

= 1 - S

Therefore, 2S = 1 and S = 1/2

If you are subtracting what S is equal to from 1, then the left side needs to be subtracted from 1 as well. You did this on the right side without doing it on the left side. The formula should read:

1-S = 1-(1 - 1 + 1 - 1 + 1 - 1 + .....)

Therefore once we remove the 1, it becomes:

-S = -(1 - 1 + 1 - 1 + 1 - 1 + .....)

Which again, makes S equal 1 or 0. Not 1/2

You might want to rethink your logic again.

Remember that the terms go on forever. It's not your normal logical adding the same thing to the left and right sides of the equation.

In fact, I did not add anything at all to the left nor to the right side of the equation.

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

The rules of maths breakdown when you mess with infinity. That's probably what's happening here. I think this is only confined to physics.

Plus, 0.9999... is equal to 1, it's mathematical fact. Maybe, I'm missing the point in what you guys are saying?

We need Babun in this thread

Plus, 0.9999... is equal to 1, it's mathematical fact. Maybe, I'm missing the point in what you guys are saying?

We need Babun in this thread

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

So what about what I am saying is wrong? The answer cannot be 1/2; it can only be 1 or 0. 1/2 is just the 'best' answer since we could never possibly know the actual answer.@Dutti wrote:You might want to rethink your logic again.

Remember that the terms go on forever. It's not your normal logical adding the same thing to the left and right sides of the equation.

In fact, I did not add anything at all to the left nor to the right side of the equation.

And okay if you didn't change anything on either side, then how did 'S = 1 - 1 + 1 - 1 +...' (which is S = S) become 'S = 1 - S'? You obviously did add a 1- to the right side of the equation and didn't do it on the left, which changes whatever the answer on the right side would have been (which cannot be determined) by a 1-.

@Dutti wrote:S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 +......

= 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)

= 1 - S

Therefore, 2S = 1 and S = 1/2

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

I agree with Eman. I can't understand why it's 1/2. It can only be 1 or 0.

We are ASSUMING it to be 1/2, right?

We are ASSUMING it to be 1/2, right?

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

one of my teachers in high school has a medal and PhD because he proved that 2+2 = 5

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

Approximation methods are always kinda flawed and results in weird equations.

s=1-1+1-1+1-1+....

is not equal to S=1-S

s=1-1+1-1+1-1+....

is not equal to S=1-S

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

@Eman wrote:So what about what I am saying is wrong? The answer cannot be 1/2; it can only be 1 or 0. 1/2 is just the 'best' answer since we could never possibly know the actual answer.@Dutti wrote:You might want to rethink your logic again.

Remember that the terms go on forever. It's not your normal logical adding the same thing to the left and right sides of the equation.

In fact, I did not add anything at all to the left nor to the right side of the equation.

And okay if you didn't change anything on either side, then how did 'S = 1 - 1 + 1 - 1 +...' (which is S = S) become 'S = 1 - S'? You obviously did add a 1- to the right side of the equation and didn't do it on the left, which changes whatever the answer on the right side would have been (which cannot be determined) by a 1-.

= 1 - (1 - 1 + 1 - 1 + 1 - 1 + .....)

= 1 - S

Therefore, 2S = 1 and S = 1/2

I already told you. The key thing to realize is that there are infinitely many terms.

So what I did was that starting from the second term (which is -1), I grouped all the terms that come after it leaving the first +1 out of the brackets:

S = 1 + (-1 + 1 - 1 + 1 - 1 + ...)

Then, I factored out the common multiplier negative one:

S = 1 - (1 - 1 + 1 - 1 + 1 -1 +...)

As you can see there is nothing added into the equation. It's just a factorization.

I also agree that 1 and 0 are possible, but I also consider 1/2 to be possible.

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

Yeah, Dutti is right. You have to remember S is infinitely many terms long, so S can be within itself, which is the confusing part.

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

i'm student at mathematics in Romania, if my math analysis teacher would see that he would get crazy

the proof is wrong because it uses false assumptions... That infinite sum (1-1+1-1...) makes 1/2 in his opinion.. in Fact that sum has no result... it is a series with the general with the general number Xn=(-1)^(n+1), which is not a convergent series... because you can take two substrings of the partial sum Sn= that have different limits (1 or 0), so the series not being convergent it can't have the sum a float number. Read some theory about series and you'll understand easy... If this result would be true all the convergence theories and math analysis would be fcked ...

you can obtain sh*ts like this if you make false assumptions (like dividing an expression with 0)

Sorry for my english...

the proof is wrong because it uses false assumptions... That infinite sum (1-1+1-1...) makes 1/2 in his opinion.. in Fact that sum has no result... it is a series with the general with the general number Xn=(-1)^(n+1), which is not a convergent series... because you can take two substrings of the partial sum Sn= that have different limits (1 or 0), so the series not being convergent it can't have the sum a float number. Read some theory about series and you'll understand easy... If this result would be true all the convergence theories and math analysis would be fcked ...

you can obtain sh*ts like this if you make false assumptions (like dividing an expression with 0)

Sorry for my english...

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

Let me give you an example of an infinite sum, that actually has a result ... An particular example of geometric series:

http://i55.servimg.com/u/f55/17/41/75/32/math210.png

The result of that sum is 1.. Isn't that beautiful?

How it translates: If you put half of a bread, and than add another quarter on it, and than 1/8 bread ....etc you'll make 1 bread

http://i55.servimg.com/u/f55/17/41/75/32/math210.png

The result of that sum is 1.. Isn't that beautiful?

How it translates: If you put half of a bread, and than add another quarter on it, and than 1/8 bread ....etc you'll make 1 bread

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

@Ion Creanga wrote:i'm student at mathematics in Romania, if my math analysis teacher would see that he would get crazy

the proof is wrong because it uses false assumptions... That infinite sum (1-1+1-1...) makes 1/2 in his opinion.. in Fact that sum has no result... it is a series with the general with the general number Xn=(-1)^(n+1), which is not a convergent series... because you can take two substrings of the partial sum Sn= that have different limits (1 or 0), so the series not being convergent it can't have the sum a float number. Read some theory about series and you'll understand easy... If this result would be true all the convergence theories and math analysis would be fcked ...

you can obtain sh*ts like this if you make false assumptions (like dividing an expression with 0)

Sorry for my english...

Using the definition of limit is no different from finding the sum of the series by adding term by term. So it is obvious that the series is divergent in that case.

But I read somewhere that there is a type of Mathematics that allows you to define the sum treating it as "what if the sum does exist then what should it be?".

Although I don't remember the name of that type of Maths, but that is what exactly being applied in the proof that leads to the sum being 1/2.

After all, since you are a student of Maths, don't you think it is troublesome to define the alternating series equal to some S? To me that "S = " is troublesome; so applying the Maths that I just mentioned treats the proof as a "what if" case. It's like treating i = sqrt(-1).

There is a little bit of insight to that type of Maths explained by Dr James Grimes (a Ph.D, btw) in one of Numberphile's videos. In it, he explains that there is a type of sum that is almost like a limit but nevertheless NOT the same as limit. He calls it "pseudo-limit" and that is what, as he explains, used to proof that the sum is 1/2.

Also, to justify that the answer is 1/2 in the REAL world (instead of pure maths), consider what would happen to the proof in the OP if the sum = 1/2 were completely false. Quantum Mechanics would fall apart because the proof that the infinite series (in the OP) = -1/12 depends on the assumption that the sum of (-1)^(n+1) = 1/2

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

I used the definition of limit just to proof that the partial sum string is not a convergent string, therefore the series can't have a result (not sure if you studied series in university to know what partial sum is) ... You can prove that this alternating series is not convergent in many ways, i gave one...@Dutti wrote:@Ion Creanga wrote:i'm student at mathematics in Romania, if my math analysis teacher would see that he would get crazy

the proof is wrong because it uses false assumptions... That infinite sum (1-1+1-1...) makes 1/2 in his opinion.. in Fact that sum has no result... it is a series with the general with the general number Xn=(-1)^(n+1), which is not a convergent series... because you can take two substrings of the partial sum Sn= that have different limits (1 or 0), so the series not being convergent it can't have the sum a float number. Read some theory about series and you'll understand easy... If this result would be true all the convergence theories and math analysis would be fcked ...

you can obtain sh*ts like this if you make false assumptions (like dividing an expression with 0)

Sorry for my english...Using the definition of limit is no different from finding the sum of the series by adding term by term. So it is obvious that the series is divergent in that case.

But I read somewhere that there is a type of Mathematics that allows you to define the sum treating it as "what if the sum does exist then what should it be?".

Although I don't remember the name of that type of Maths, but that is what exactly being applied in the proof that leads to the sum being 1/2.

After all, since you are a student of Maths, don't you think it is troublesome to define the alternating series equal to some S? To me that "S = " is troublesome; so applying the Maths that I just mentioned treats the proof as a "what if" case. It's like treating i = sqrt(-1).

There is a little bit of insight to that type of Maths explained by Dr James Grimes (a Ph.D, btw) in one of Numberphile's videos. In it, he explains that there is a type of sum that is almost like a limit but nevertheless NOT the same as limit. He calls it "pseudo-limit" and that is what, as he explains, used to proof that the sum is 1/2.

Also, to justify that the answer is 1/2 in the REAL world (instead of pure maths), consider what would happen to the proof in the OP if the sum = 1/2 were completely false. Quantum Mechanics would fall apart because the proof that the infinite series (in the OP) = -1/12 depends on the assumption that the sum of (-1)^(n+1) = 1/2

Regarding the S, like you said it's wrong to assume that S (real number) is the result of that infinite sum and make operations with S, because S does not exist ...

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## Re: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + ........ = -1/12

@Ion Creanga wrote:I used the definition of limit just to proof that the partial sum string is not a convergent string, therefore the series can't have a result (not sure if you studied series in university to know what partial sum is) ... You can prove that this alternating series is not convergent in many ways, i gave one...@Dutti wrote:@Ion Creanga wrote:i'm student at mathematics in Romania, if my math analysis teacher would see that he would get crazy

the proof is wrong because it uses false assumptions... That infinite sum (1-1+1-1...) makes 1/2 in his opinion.. in Fact that sum has no result... it is a series with the general with the general number Xn=(-1)^(n+1), which is not a convergent series... because you can take two substrings of the partial sum Sn= that have different limits (1 or 0), so the series not being convergent it can't have the sum a float number. Read some theory about series and you'll understand easy... If this result would be true all the convergence theories and math analysis would be fcked ...

you can obtain sh*ts like this if you make false assumptions (like dividing an expression with 0)

Sorry for my english...

Using the definition of limit is no different from finding the sum of the series by adding term by term. So it is obvious that the series is divergent in that case.

But I read somewhere that there is a type of Mathematics that allows you to define the sum treating it as "what if the sum does exist then what should it be?".

Although I don't remember the name of that type of Maths, but that is what exactly being applied in the proof that leads to the sum being 1/2.

After all, since you are a student of Maths, don't you think it is troublesome to define the alternating series equal to some S? To me that "S = " is troublesome; so applying the Maths that I just mentioned treats the proof as a "what if" case. It's like treating i = sqrt(-1).

There is a little bit of insight to that type of Maths explained by Dr James Grimes (a Ph.D, btw) in one of Numberphile's videos. In it, he explains that there is a type of sum that is almost like a limit but nevertheless NOT the same as limit. He calls it "pseudo-limit" and that is what, as he explains, used to proof that the sum is 1/2.

Also, to justify that the answer is 1/2 in the REAL world (instead of pure maths), consider what would happen to the proof in the OP if the sum = 1/2 were completely false. Quantum Mechanics would fall apart because the proof that the infinite series (in the OP) = -1/12 depends on the assumption that the sum of (-1)^(n+1) = 1/2Regarding the S, like you said it's wrong to assume that S (real number) is the result of that infinite sum and make operations with S, because S does not exist ...

lim of S_n as n approaches infinity...that's pretty basic; treating the partial sums as an infinite sequence that is.

You don't need a Math major to understand that since it's learned in first year anyway. Even engineering students understand that.

However, most people do not realize that the limit definition of partial sums taking n to infinity is the same as adding the terms in the series itself one by one, and I'm not sure you've realized this.

Regarding the bold sentence I did not claim that. I guess you misunderstood my saying "what if it DOES exist". Regardless of what you understand, you seem to have overlooked Dr Grimes's explanation. That's pretty bold.

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