I would try
put 1 on top, and distribute the
exp(-Y[t])
into each term of the sumsimplify each term of the sum, by summing the exponents
or some such algebra.
Another way to look at it ...
You say the overall fraction is between 0 and 1. Well, if it's close to 1, that means the denominator is bound to be reasonable.
If it's close to 0, that means the denominator is very large. So if you're summing the denominator, and it gets unreasonably large, just quit. You know it can only increase.
Also, you could sort the terms of the sum, doing the ones with the largest exponent first. That way you get to the unreasonableness sooner.