An infinite amount of permutations,

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  • #1
daniel_i_l
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an "infinite" amount of permutations,

Look at how the sum of the alternating harmonic series can be changed by rearranging the terms:
http://mathworld.wolfram.com/RiemannSeriesTheorem.html
But doesn't this involve a non-finite amount of permutations? If this counts as a rearrangement then why can't we change 0,1,0,1... into 0,0,0... by the following argument:
If we know that for a sequence a_n, for every N>0 for every n<N a_n = 0 then obviously
a_n = 0,0,0... right?
But suppose we take the sequence b_n = 1,0,1,0... Then with N permutations (it doesn't really matter how many) we can rearrange it so that the first N elements are 0. So with an "infinite" amount of permutations we can make the first N elements 0 for all N!
I'm pretty sure that the problem here is in the expression "infinite amount of permutations" but I can't see any way to describe the rearrangement on wolframs site. Can someone help me clear this up?
Thanks.
 

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  • #2
HallsofIvy
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Look at how the sum of the alternating harmonic series can be changed by rearranging the terms:
http://mathworld.wolfram.com/RiemannSeriesTheorem.html
But doesn't this involve a non-finite amount of permutations? If this counts as a rearrangement then why can't we change 0,1,0,1... into 0,0,0... by the following argument:
If we know that for a sequence a_n, for every N>0 for every n<N a_n = 0 then obviously
a_n = 0,0,0... right?
But suppose we take the sequence b_n = 1,0,1,0... Then with N permutations (it doesn't really matter how many) we can rearrange it so that the first N elements are 0. So with an "infinite" amount of permutations we can make the first N elements 0 for all N!
That does not follow. For one thing you haven't defined what you mean by an "infinite" amount of permutations. For another, you cannot simply assert that something that is true "for finite n" becomes true for an "infinity"- whatever you choose to mean by that.

I'm pretty sure that the problem here is in the expression "infinite amount of permutations" but I can't see any way to describe the rearrangement on wolframs site. Can someone help me clear this up?
Thanks.
I, on the other hand, don't see how that at all describes it! That site is talking about all possible rearrangements- some of which might involve on a single swap of two numbers. You might well talk about "an infinite number of permutations" but that means, as is meant by the site, different results for an infinite number of different "regular permutations"- not a single permutation that involves an infinite number of swaps- which is what you appear to be thinking.
 
  • #3
daniel_i_l
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First of all, "permutations" was the wrong word - I should have said "swaps". Where by a "swap" I mean switching between two elements.
Let's say that you rearrange the sequence a_n to sequence b_n so that
[tex]b_{2n} = a_{2n+1}[/tex] and [tex]b_{2n+1} = a_{2n}[/tex]for all n. Isn't this the same as performing an "infinite" amount of swaps on a_n (1-2, 3-4, 5-6 ...)? If so, why can't you use the an "infinite" amount of swaps on 0,1,0,1... to turn it into 0,0,0...?
I don't know how to define this "infinite" amount of swaps and this is the problem - but then what does it mean in the case of the a_n -> b_n rearrangement which clearly is well defined?
Thanks.
 

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