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?  

I would argue that the most important reason for being concerned about convergence in the 'real world' is that statements proved in the absence of concerns about convergence can be outright false!  The simplest example that comes to mind is the geometric series; the fact that $\sum_{n=0}^{\infty}x^n = \frac{1}{1-x}$ is incredibly useful and has plenty of applications, both directly to the real world and in doing other mathematics that then gets applied to real-world problems — but you have to be careful not to conclude that $1+2+4+8+\cdots = -1$ from it!
(and I'm well aware that even this 'absurd' conclusion can make sense in certain circumstances — but it's not true in $\mathbb{R}$ as it stands, and there are much more insidious versions of the same error where the interpretations that can be applied here make much less sense.)
