Are questions of convergence important in real life?

Question

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?

Answer 1

Whenever you use a numerical method to approximate something, you'd like to know that your numerical answer will be close to the actual value. A common situation is that the numerical approximation is $A(n)$ where $n$ is a parameter (e.g. the number of steps that are used). If the true answer is $T$, you'd like to know that $\lim_{n \to \infty} A(n) = T$, which says that you can ensure that your approximation is as close as desired to the true answer by taking $n$ large enough.

Of course you'd really like to have more detailed information (i.e. for a given tolerance $\epsilon$, how large to take $n$ in order to have $|A(n) - T| < \epsilon$), but the fact that the limit is $T$ is a good start - if it was not true, it would mean that if you want really good approximations you should look for different methods.

Answer 2

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.

Answer 3

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.)