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        riddles >> medium >> lim and lim
        
(Message started by: Mickey1 on Feb 22^nd, 2011, 7:34am)

Title: lim and lim
Post by Mickey1 on Feb 22^nd, 2011, 7:34am

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?

Title: Re: lim and lim
Post by towr on Feb 22^nd, 2011, 10:25am

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.