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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?

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what's real life? –  tentaclenorm May 14 '12 at 22:59
Whatever the "finitude of the universe" means, we often deal with mathematical models of the universe that are not finitary (e.g. differential equations), and so it's natural to expect that analytic issues are relevant to these models. For example, lack of convergence in complex analysis signals the presence of poles, which are important in e.g. control theory (en.wikipedia.org/wiki/Control_theory). –  Qiaochu Yuan May 14 '12 at 23:07
Questions of convergence were originally considered in connection with solutions of differential equations, because people needed to solve practical problems of physics and engineering. –  MJD May 14 '12 at 23:56
My (admittedly extremely cursory) understanding of the state of things is that a major open problem in mathematical physics is understanding "Feynmann integrals", which are morally the right thing to study but nevertheless diverge wildly. Convergence is the entire issue here! –  Aaron Mazel-Gee May 15 '12 at 1:08
@AaronMazel-Gee: This is another good example of the irrelevance of convergence. Arguably, quantum electrodynamics is the most precise physical model of nature. Every day physicists compute Feynman integrals without worrying about the convergence issues you refer to. –  user26872 May 15 '12 at 20:19

3 Answers 3

up vote 9 down vote accepted

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.

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Right. E.g. if you want to use a differential equation to simulate a system, you'd like to be able to use Euler's method or something similar to approximate the solutions, and you'd like to be able to guarantee that your approximations will converge to the actual behavior predicted by the DE... –  Qiaochu Yuan May 15 '12 at 0:16
+1 though in "real world" differential equations to simulate systems may not even be valid given that functions in the "real world" may not even be $C^0$. –  user17762 May 15 '12 at 0:54
@Marvis: All we can ever do is study a mathematical model of the "real world", not the "real world" itself, which is far too complicated. We hope that the model mirrors in some way those features of the real world that are important in the given context. And we are often pretty successful at that - we make bridges that don't fall down and machines that work. –  Robert Israel May 15 '12 at 1:42
So the differential equation is part of the model, not of the "real world". –  Robert Israel May 15 '12 at 1:47

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.

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If it helps, the convergence happens in finite time –  Alex R. May 14 '12 at 23:28
This is actually a great example of the irrelevance of convergence; we didn't stop being capable of movement just because nobody knew how to resolve Zeno's paradox (and whether the notion of convergence counts as a resolution of Zeno's paradox is debatable). –  Qiaochu Yuan May 14 '12 at 23:28

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.)

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