I'm looking for interesting paradoxes involving infinity.
One example I thought of last night:
[math]\sum_{k=1}^{\infty}k  \sum_{k=1}^{\infty}2k1 = 2\sum_{k=1}^{\infty}k[/math]
[math]\left(1 + 2 + 3 + 4 + \cdots\right)  \left(1 + 3 + 5 + 7 + \cdots\right) = 2 + 4 + 6 + 8 + \cdots[/math]
In other words, if you subtract all the odd terms from the series, that's the same as doubling all the terms.
As I understand it, this is roughly equivalent to the BanachTarski paradox
EDIT: sorry, I failed at basic arithmetic... This doesn't work with the harmonic series
Infinity and Paradoxes
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Infinity and Paradoxes
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Re: Infinity and Paradoxes
The difference is anything but well defined, it depends massivly on how you put your brackets. Conversly, I could argue that each number in the second series is at least as big as the corresponding one in the first sum, so I get
[math](11)+(23)+(35)+(47)+...= \sum_{k=0}^\infty k =  \infty[/math]
The issue is that the difference of these two series is not well defined, depending on just how I construct the differences of partial sums I can get very different results in the limit, and there is most certainly no "one correct" way to do it.
This is in no shape or farm related to the Banach Tarski Paradoxon at all, it's just an ill defined term (and therefore a nonsensical conclusion).
[math](11)+(23)+(35)+(47)+...= \sum_{k=0}^\infty k =  \infty[/math]
The issue is that the difference of these two series is not well defined, depending on just how I construct the differences of partial sums I can get very different results in the limit, and there is most certainly no "one correct" way to do it.
This is in no shape or farm related to the Banach Tarski Paradoxon at all, it's just an ill defined term (and therefore a nonsensical conclusion).
Re: Infinity and Paradoxes
Yeah, I wouldn't say that this is roughly equivalent to the BanachTarski paradox... however, I would say that they have in common the idea of paradoxical decompositions of "infinity". At any rate, if you liked what you came up with, you would probably really be blown away by what Riemann had to say about bad series:
http://en.wikipedia.org/wiki/Riemann_series_theorem
It basically says that if a series converges, but that's because of cancellation (that is, the sum of the absolute values of the terms goes to infinity), then if you give me any real number, I can rearrange the terms of the original series to add up to that number. I can also rearrange them to go to infinity or negative infinity. It's a pretty strong indication about how commutativity breaks down under taking limits.
If you want more stuff on infinity and its paradoxes, there are plenty of interesting philosophical and mathematical questions floating around on it, since we've been collectively wrestling with the concept since its inception. If you haven't already, you might also want to read some set theory and learn about cardinality, where you'll see some statements about how there are different "sizes" of infinity. An "infinite" number of different sizes, of course.
http://en.wikipedia.org/wiki/Riemann_series_theorem
It basically says that if a series converges, but that's because of cancellation (that is, the sum of the absolute values of the terms goes to infinity), then if you give me any real number, I can rearrange the terms of the original series to add up to that number. I can also rearrange them to go to infinity or negative infinity. It's a pretty strong indication about how commutativity breaks down under taking limits.
If you want more stuff on infinity and its paradoxes, there are plenty of interesting philosophical and mathematical questions floating around on it, since we've been collectively wrestling with the concept since its inception. If you haven't already, you might also want to read some set theory and learn about cardinality, where you'll see some statements about how there are different "sizes" of infinity. An "infinite" number of different sizes, of course.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.
Re: Infinity and Paradoxes
What you've shown is manipulation of symbols without meaning. A divergent sum doesn't actually represent a value, so how can you subtract them? Just another example of how intuition seems to fail us when dealing with infinities.
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 doogly
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Re: Infinity and Paradoxes
Here's your problem, you forgot that
[math]\sum_n n = \frac{1}{12}[/math]
From there it should be easy to see where to go.
[math]\sum_n n = \frac{1}{12}[/math]
From there it should be easy to see where to go.
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Keep waggling your butt brows Brothers.
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 ImTestingSleeping
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Re: Infinity and Paradoxes
You can't rearrange a series which isn't absolutely convergent.
If you are looking for something strange, then I would offer the fact that for any conditionally convergent series [imath]\sum a_n[/imath], for any number x there exists a rearrangement of [imath]\sum a_n[/imath] such that [imath]\sum a_n[/imath]=x.
[Edit] Whoops, didn't notice z4lis already linked Riemann's theorem in an above post. Neat stuff.
[Edit] Whoops, didn't notice z4lis already linked Riemann's theorem in an above post. Neat stuff.
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