1=2 Again

[Good Dreams]

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

i't equal to:

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

and equal to:

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

In the first one as (1-1)=0 then the all thing is equal to 1

But in the second one as (1-1)=0 then the all thing is equal to 2

How to explain this? :D

[parksie]

Order of precedence. Addition and subtraction are equal, but brackets are higher. You're forcing the operations to take place in a different order :)

[Guv]

The real problem is that you are doing illegal, immoral, and fattening mathematical operations.

In most contexts, operations which cannot be defined in some finite fashion are either not defined or not allowed.

The infinite series you are using does not converge. Its sum is undefined.

In general, any process which requires an infinite number of operations is undefined, illegal, invalid, immoral, whatever.

Convergent series (for example) are defined in a manner which avoids a requirement for an infinite number of operations to determine the limit and/or apply the definition.

[simonm]

Rather than say the answer is undefined, what about one of the following possibilities:


The sum of the series is equal to both 1 and 2 simultaneously?
The sum of the series is equal to 1.5 (an average)?
The sum of the series is equal to 0 (1+1+1+1... -1-1-1-1...)?

[Guv]

Simonm: You gave four plausible values for the sum of that series.

While I do not like the average much, it is not unreasonable. BTW: I always saw this series presented as 1 -1 + 1 -1. . . which has an average of 1/2.

I object to saying 1 and 2 simultaneously, but either value looks reasonable by itself.

What you have done is present a wonderful argument for saying that the sum is undefined. Id est: It is impossible to decide which of several possible values is correct. That situation is one meaning of undefined.

[simonm]

I object to saying 1 and 2 simultaneously, but either value looks reasonable by itself.

What reason is there to choose either value individually over the other? I would prefer to take both values simultaneously rather than one or the other.

[Good Dreams]

I didn't want to be immoral.
I just think that when infinite is used, the rules of the game are diferent... ;)

Still, some of them persist like: infinite^-1=0

Hey, I'm only a 11th grade student, it's possible for me to be wrong. :p

[Guv]

Simomn: Would you be willing to accept the following two statements? The sum of the series is two. The sum of the series is not two.How about the following? Simonm: You have been found guilty of murder in the first degree for strangling your enemy. I want to appeal. What about the three unimpeachable witnesses who testified that I was in New York when the murder was committed in London? What about the credit card receipts from the New York restaurants? Appeals judges reply: We are willing to simultaneously believe that you committed the murder and that you were 3000 miles away at the time it was committed. You were in London that night and you were not in London that night. We ignore picky logical concepts about a statement and its converse not being simultaneously true.Goodreams: Glad that you do not want to be immoral. I hope you do not want to be unethical either.

You are correct, the rules involving infinity are different, but there are still some rules.

[simonm]

The idea of the sum of the series having more than one value goes against our ideas of mathematical and logical consistancy.

But, Guv, you yourself said:

...the rules involving infinity are different, but there are still some rules.

Perhaps it is possible to conceive that a non convergent infinte series might yield more than one result?

The infinite series you are using does not converge. Its sum is undefined.

The assumption that leads you from the first statement 'The infinite series you are using does not converge' to the following statement 'Its sum is undefined' should be examined.

Because there is more then one (possible) result we say it is undefined. Is that a fair conclusion?

[L0phtpDK]

Why in the world would u need this?

[simonm]

Need an infinite series perhaps?

You don't need it.

We don't need it but some of us want it.

Others of us don't believe we can have it even if we want it.

None of us agree what it actually is but we like talking about it.


Hope this helps? :)

[Good Dreams]

Originally posted by simonm
None of us agree what it actually is but we like talking about it.


I agree 100% with that :D

[Guv]

Simonm: If you do not like undefined for the sum of a nonconvergent infinite series, what term would you prefer? What about one of the following? The sum is undefined. This is included to provide a complete list of possibilities. The sum is not known. The sum cannot be assigned a value. The sum of the series after some number of terms is whatever. Id est: Do not refer to the sum without specifying a finite number of terms. The sum is illegal, immoral, and fattening. The sum is 1 on Tuesdays, Thursdays, and Saturdays; 2 on other days of the week, expect that for all of February we defer to Guv and call it undefined. The series is nonconvergent, and we do not talk about the sum. The sum should not be mentioned in polite company.BTW: Guv;s birthday is in February.

[simonm]

Is that becuase sum of an infintie series that does not converge seems to have multiple answers does not imply that all answers are therefore invalid.

Infinite sets display many characteristics that are strange and counter-intuitive to us with our understandings of finite sets. Perhaps this is just another strange property of infinite sets that we have yet to get used to.

[Good Dreams]

What about the case of 0/0 ?

Considering a line:
y = mx + b

we can say that m = (y-y0)/(x-x0).
Let y0 = 0 and x0 = 0

when x=2 and y=2, m = 1

But when x=0 and y=0? It stays 0/0

However:
1. anynumber / 0 = infinite
2. m is constant

So, as the constant value is 1, 0/0 = 1 in this case.
But the line could have a diferent value for m and 0/0 would have a diferent value.
I suppose it is correct to say that infinite can be different values... or simply undefined. :rolleyes:
Correct me if I'm wrong or if I'm being immoral and unethical. :D

[Larry Mayfield]

The assumption made is that these are the identical series. I propose that they cannot be. The first is:
1+1-1+1-1+1-1+1-1...
The 2nd series:
1+(1-1)+(1-1)+(1-1)...
shows that the elements of the series are
(1-1) added continuously. (after the constant 1 of course).
The 3rd Series has a constant 1 added to a constant 1 and then the elements are (-1+1) added continuously.
If the reordering of the parenthesis were allowed in this case then a consequence of this reordering would be that 1 not only equals 2, but also 3, through infinity. In other words you could change the 3rd series from:
1+1+(-1+1)+(-1+1)+(-1+1)... to:
1+1+(1-1)+(1-1)+(1-1)...
This would allow you to start all over again. I believe this shows that using the parenthesis in this way must be a violation of a rule that should be applied in this case.
Either the series are not the same, and then of course one series can equal 1 while the other equals 2 or the rules of series manipulation must have been violated because you can't have:
1+(1-1)+(1-1)+(1-1)... = 1+1+(1-1)+(1-1)+(1-1)...

1+1+(-1-1) the answer is 0

1+1-(-1-1) the answer is -4

1+1+(-1+1) the answer is 2

1+1-(-1+1) the answer is 2

1-1+(-1-1) the answer is -2

1-1-(-1-1) the answer is -2

1-1+(-1+1) the answer is 0

1-1-(-1+1) the answer is 0

And even if they were in infinity, the answer would still be the same...

Hope this clarifies a few things.. :)

Knight Vision

[simonm]

No matter where you put the brackets around the original series, for a specified finite number of items, the sum is always the same.

It's only when you consider all elements in the infintie series that these contradictions seem to arise.

[Guv]

Simonm: Your last post got it right.