- #1
daniel_i_l
Gold Member
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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.
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.