Re: Function on an infinite number of real variables

On Fri, 05 Oct 2007 00:59:08 -0000, Paul Smith <phhs80@xxxxxxxxx>
wrote:

An example could be:

maximize sum( (a^(t-1)) * ln(x_t) ) with t (discrete) from 1 to +oo,

such that the sum of all x_t's is equal to a constant k. Assume that 0
< a < 1.

To me, the most natural approach would be to solve the n-dimensional
version, 1 <= t <= n, and see how the solutions (both the x's and the
maximum sum) progress as n increases. You may need to do this
numerically. Perhaps the solution is unique, and hopefully, there will
be a clear pattern for the structure of the solutions. Once you
discover the pattern, even though discovered experimentally, it may be
possible to prove that it holds for all n (perhaps by induction).
Finally, try to show that the solution for infinitely many variables
is necessarily a limit (in some sense), as n approaches infinity, of
the n-dimensional solutions.

quasi
.

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