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Topic: lim and lim (Read 1434 times) |
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Mickey1
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lim and lim
« on: Feb 22nd, 2011, 7:34am » |
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I am wondering about the transition from S(n) to Lim S(n) when n goes to infinity
Take for instance the sum over all n of 1/n^2
Compare it to
Riemann’s-Zeta(s)*product-over-all-primes-p (1-1/p^s)=1
In the first case it is obvious how this can be done.
In the second case it is also obvious what is meant but if you look at it from an engineer’s (or a practical person’s) point of view, the primes are increasingly difficult to find and the actual summation would never be done, and I assume that for some s near 1 the error would be arbitrarily high for any attempt i.e. for any sequence from 2 to n.
The second seems to me to be perhaps "legal" but still questionable.
Don’t you think there is some kind of difference between the two alternatives?
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towr
wu::riddles Moderator
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Re: lim and lim
« Reply #1 on: Feb 22nd, 2011, 10:25am » |
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I don't think there is a very big difference. In both case you have the problem that you can't perform an infinite number of operations. The first may be easier to approximate, but that doesn't really change the problem.
http://mathworld.wolfram.com/EulerProduct.html shows how you can manipulate parts of the second formula to get around the problem of infinity.
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