Nope. You're thinking about this mathematically, not algorithmically. The question is essentially to decide whether the output of a TM is all zeros or whether it will eventually output a 1. If it outputs a 1 you will know that in a finite amount of time by running the program, but if it never outputs a 1 then you can't tell that by running the program, and it's impossible in general to tell any other way because that would be tantamount to solving the halting problem.

Here is a concrete example: write a program that systematically tests all numbers for counterexamples to the Goldback conjecture. Every time a number fails to be a counterexample, output a zero, otherwise output a 1. Consider the output of that program to be a real number expressed as a binary decimal. If that number is not zero you will know in a finite amount of time, but if it is zero you will not know that until someone proves the Goldbach conjecture.