Are questions of convergence important in real life? In the real world, do we ever need to worry about convergence and what not? I am not talking about whether recursive functions and such terminate, but convergence in analysis. It seems like the finitude of the universe makes questions like that meaningless. I ask because it often seems like physicists and statisticians are very lax about convergence. I know physicists might seem to care about it every once and a while (wave functions must be in normalizable i.e. in $L^2$) but it doesn't appear to be truly important. 
So what are some real world reasons for concerning ourselves with convergence?  

One funny example I can think of right of the top of my head is Zeno's paradox of movement
To move on mile, you first would have to move half a mile, and then half of that half, and so on and so on, so it seems you never actually reach the mile....We now can say that converges to 1 mile, but back then that was a real brain exploder.
