# How to think when devising a TM that decides if a DFA rejects some string?

The title should explain my question thoroughly enough. I can't seem to get started anywhere. Intuitively it seems like some kind of brute-forcing would work i.e if the DFA has the symbols $$\Sigma$$ you would simply be passing strings from $$\Sigma^*$$ until the DFA rejects. What would even be the time complexity of this though? Or am I completely out and about?

The problem with your approach is that there are infinitely many strings in $$\Sigma^*$$. If the automaton keeps accepting and accepting, how would you know hen to stop?