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?