*Originally posted by FabianFnas*

**In the original Youtube clip they said that 1-1+1-1+... can be either plus one or zero. So 'we set it to be the average = 1/2'. This is BS if anyone ask me. One cannot just reason what the sum is. And this result he uses in the proof of sum of positive integers. Very sloppy, indeed it is.
**

But in this video https://www.youtube.com/watch?v=PCu_BNNI5x4 he ...[text shortened]... s off track. Where is the error in his reasoning? It must be there somewhere! Where is the flaw?

YouTube
I think that said 'proof' the infinite sum of that infinite series adding to 1/2 i.e.

1 - 1 + 1 - 1 + 1 - 1 ... = 1/2

is erroneous.

Lets go through his reasoning one step at a time;

He first says the infinite series;

1 - 1 + 1 - 1 + 1 - 1 ...

can be rewritten with an infinite series of added brackets as;

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

with what is in the brackets canceling down to 0 thus;

(0) + (0) + (0) ... = 0

and so he says that equals 0 'thus' we have;

1 - 1 + 1 - 1 + 1 - 1 ... = 0

But then he says that infinite series can be rewritten with an infinite series of added brackets as;

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

with what is in the brackets canceling down to 0 thus;

1 + (0) + (0) + (0) ... = 1 + 0 = 1

and so he says that equals 1 'thus' we have;

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

OK, already I think we have a big problem! For we now have that very SAME infinite series apparently equaling two different unequal numbers depending on how you completely arbitrary place the brackets. So you can now write the infinite series on both sides of the equal sign thus;

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

which can be arbitrarily rewritten with an infinite series of added brackets differently on either side of that equals sign as;

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

thus, using the previous results, that simplifies to;

0 = 1 Contradiction! 0 does not equal 1 !

I think this contradiction should tell us that the sum of that infinite series is not a definable number!

OK, but lets ignore that contradiction (like he did) and continue regardless!

He then says let;

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

This is implicitly assuming that you can validly write that as S which in turn implicitly assuming that that infinite series is a definable number, which it isn't.

he then posses the question what is 1 - S equal to?

He then writes this out as;

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

But then says that of you remove the brackets you must reverse the signs of the 1s in the brackets thus we get;

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

and then says the RHS is the SAME infinite series we started with 'thus' we have;

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

and that implies 1 - S = S which implies 1 = 2S which implies S = 1/2.

BUT, and here is the logical error, it being the SAME infinite series we started with doesn't mean it has the SAME value we started with! That is because we had previously given the results of BOTH;

1 - 1 + 1 - 1 + 1 - 1 ... = 0

AND

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

Thus using those two results you could just as easily write BOTH;

1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ... ) = 1 - (0) = 1

AND

1 - S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ... ) = 1 - (1) = 0

So now we have yet additional contradictions of;

1 - 1 + 1 - 1 + 1 - 1 ... = 0

AND

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

AND

1 - 1 + 1 - 1 + 1 - 1 ... = 1/2

ALL three of the above contradict each other thus confirming the sum of that infinite series is not a definable number!

OK but now he says one method to get the sum of a DIFFERENT infinite series

1 + 1/2 + 1/4 + 1/8 ...

is to get the limit of the average of the sum as you increase n towards infinity where n is the first finite n number of terms of that series.

So for n=2 that is the average of;

1 and 1/2

and for n=3 that is the average of;

1 and 1/2 and 1/4

and for n=4 that is the average of;

1 and 1/2 and 1/4 and 1/8

and so on and you can see that as n increases the average tends to 2 as required. That is fine.

But then he applies the same method to the original infinite series of;

1 - 1 + 1 - 1 + 1 - 1 ...

and notes that using the same method you get the limit of the average of the sum as you increase n towards infinity of 1/2.

BUT the difference here is while no matter how you put an infinite series of brackets around the 1 + 1/2 + 1/4 + 1/8 ... and use another valid method to find its limit you don't get contradictory results, in contrast, as previously shown, you DO get contradictory results if put an infinite series of brackets around the 1 - 1 + 1 - 1 + 1 - 1 ... (with it equaling either 0 or 1 or 1/2).

So I think just because you get a consistent result of 1/2 if you apply this other method (of limit of averages) of finding the limit accounts to nothing because you are still left with the earlier contradictory results that are contrary to it being 1/2 (because they say it is 0 and 1)

The simplest way I can say what I personally think the 'flaw' is of his reasoning is just to simply point out that he ignored the earlier contradiction of that same infinite series both equaling 1 AND zero and then continued regardless even though I think that that proves right from the start that there is no definable number as a limit thus any later argument to define a limit is irrelevant. Do you agree with that?

Well, that is what I make of it.