The end of the world...

Consider the infinite sum of 1s and -1s; that is,

 

1 - 1 + 1 - 1 + 1 - 1 + 1 - 1.......forever and ever.

 

If you group the terms in pairs, you get

 

(1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) ..........forever and ever...which is really just

0 + 0 + 0 + 0............=0

 

On the other hand...if you group the terms as follows:

 

1 + (-1 + 1) + (-1 + 1) + (-1 + 1).........thats just

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

 

0=1?!?!?!?!?! THE END OF THE WORLD HAS BEGUN!

You can't use the associative law here because the two series given converge differently. Therefore, 0 != 1.

HV speaks the truth.

I dispute your response.

 

1) The set of integers Z is a ring. That is, (Z, +), together with the binary operation * (multiplication) maintains the following 3 properties:

 

i) a * (b * c) = (a * b) * c

ii) a * (b + c) = (a * b) + (a * c)

iii) (a + b) * c = (a * c) + (b * c)

 

As such, it is easy to show that any ring (e. g. (Z, +, *)) is an abelian group with respect to addition. A group G is said to be abelian w.r.t. addition if for elements x, y in G, x + y = y + x.

 

So having established that the set of integers maintains the commutative property of addition, there is an easy argument (not referenced due to lack of notation on this board) that shows that all subgroups of abelian groups are also abelian. So, the set {-1, 1}, being a subset of Z, also maintains the commutative property.

 

In terms of adding these two elements ad infinitum, the associative law is never an issue in terms of the grouping in the original post. Any two elements may be transposed at any time thanks to the abelian nature of the set, and therefore there is no falsity in the above argument.

 

What is interesting, however, is that there is a whole school of thought that deals with alternating series (which is what is given) that converge based on commutativity or lack thereof.

 

For example, the original sum is not always convergent to the same value depending on grouping.

 

1 - 1 + 1 - 1 + 1...etc can be rewritten (thanks to commutativity) as

1 + 1 - 1 + 1 - 1 (transposing the order of every pair of elements excluding the first entry)

and can be subsequently grouped as

(1 + 1) - 1 + 1 - 1...which = 2 if taken infinitely often.

 

As an aside, there is no way to write the sum of -1s and 1s using only commutativity which does not produce a convergent series. That is; lim n-->infinity from 0 to infinity of (-1)^n is a conditionally convergent series, it just turns out that the value the series converges to varies depending on the grouping of the terms.

 

In conclusion, this is not an associative argument, nor two different series. hurray math

Edited September 17, 2007 by Hana

The thing is, you can't set those series equal to each other because they ARE different series, at least thats how I see it. The cases of (-1)^n are separated out into different series based on what they converge to...

 

The fact that those two series are not equal here is a failure of the application of the associative law. Namely, you can't put those parentheses there, and group them, and prove anything about them being equal. They are simply not the same.

 

Anyway, I'm not a math major, so tear this argument apart : P

meh they are the same, but im jsut bored and forum trolling

i agree that conditionally convergent series can be manipulated to equal whatever you want, and that you cant prove results like 0=1 based on it...it was just some silly math "joke"

 

i still

Edited September 17, 2007 by Hana

I <3 you too, math jokes are fun, and I'm pissed at this paper I have to write for a history class... engineers taking a history class! the nerve!

 

My eng. statistics prof. did one on the board I thought funny the first day of class -

 

He told us that he would prove

Sin x = 6

n

 

He proceeded to cross out both the n's, leaving...

 

Si x = 6

Uh..

 

4 (+1) and 4 (-1) = 0.

 

4(+1) and 3 (-1) = 1.

 

I'm failing to see the end of the world in these two completely different equations.

math shmath....I'm gonna go buy all the canned goods I can.

 

See ya after the apocalypse suckas!

 

/bunker

too much math, didn't read.

Think I'll gonna have to go with my Gut on this one.

 

Diet DrPepper may taste like regular DrPepper, but it still sucks!

 

Heh. 1/3 = .3333

1/3 x 3 = 3/3 = 1

.333333 x 3 = .999999

 

Lyke fvck fractions, man.

 

I thought I was going to have teacher/student sexy time with my Calc teacher, who I thought hated me. I had to stop by after class to turn in an assignment cause I missed a day, and she's all "wait here...there's something I want to show you." This is at like, 5 on a Tuesday night, I was doing debate work after school before leaving. So the school is pretty much empty, aside from two debate students, the teacher, my calc teacher, and me. o_o''

She comes back in, wearing a bath robe, and is like "Alright now, time for your lesson..."

 

...Not really. =[

But man, I was expecting it. instead she came back in and just showed me some instances where math gets funky. the 0! being one of them. But yeah.

 

x_x

Heh. 1/3 = .3333

1/3 x 3 = 3/3 = 1

.333333 x 3 = .999999

 

Lyke fvck fractions, man.

 

I thought I was going to have teacher/student sexy time with my Calc teacher, who I thought hated me. I had to stop by after class to turn in an assignment cause I missed a day, and she's all "wait here...there's something I want to show you." This is at like, 5 on a Tuesday night, I was doing debate work after school before leaving. So the school is pretty much empty, aside from two debate students, the teacher, my calc teacher, and me. o_o''

She comes back in, wearing a bath robe, and is like "Alright now, time for your lesson..."

 

...Not really. =[

But man, I was expecting it. instead she came back in and just showed me some instances where math gets funky. the 0! being one of them. But yeah.

 

x_x

 

 

that was starting to turn me on till you said it was all made up

too much math, didn't read.

 

too much math, didn't read.

 

that was starting to turn me on till you said it was all made up

 

Only the bathrobe part. =[

She was a biotch anyways.