Find the maximal value of the expression
where
are positive numbers satisfying for each
Well, here's something just to get the ball rolling...
If you use powers of 2, that is , that gives you . Because of the inequality, you can let and for . This gives
Now imagine letting be an arbitrarily large number L, and denote . Now the maximal value of would be gotten by defining , giving it a value of which tends to 1 as L grows large. Thus
I am trying to imagine an induction approach to repeat this last trick on and so on, but as of right now, the highest value I can generate is
I don't believe that the construction with L and L+c leads to any improvement on the previous total of 1004.5. In fact, if you choose then it affects the term , changing the last two terms of the sum from to which tends to 1 as L→∞. So nothing is gained.
I suspect that the value 1004.5 obtained by taking for is the best possible.
Sorry... So far the most efficient selection of a's we have been able to find is the following:
...
And so on, amounting to
This gives your sum
This is probably the maximum, as Opalg corrected my faulty logic in my last post. What we lack though, is a proof.