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We had a discussion regarding this answer on Electrical Engineering.

The answer in question discussed a finite continued fraction. We're wondering whether it's a closed-form expression or not.

According to the wikipedia on closed-form expressions, continued fractions aren't closed-form expressions but only analytical expressions, see the Comparison of different classes of expressions. However, in the text above the table, we read:

Unlike the broader analytic expressions, the closed-form expressions do not include infinite series or continued fractions; ...

So it seems to be talking only about infinite continued fractions. The text seems to suggest that finite continued fractions are closed-form expressions, but this contradicts the table.

Are finite continued fractions closed-form expressions, and if not, why not?

Of course, at least some finite continued fractions are closed-form expressions, for example $\frac{1}{2}$. But are all finite continued fractions (and in particular the one from the answer on EE.SE) closed-form expressions?

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The non-closed-formedness of that answer is that the size of the given expression depends on the size of the problem. That's quintessentially not a closed form, just as much as something with, say, $\sum_{k=1}^n$. The fact that it's finite for any particular invocation is completely beside the point.

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  • $\begingroup$ Okay, thank you, that's good to know. But a finite continued fraction that is defined 'without dots' (I'm not a mathematician, I wouldn't know how to express this exactly) would be a closed-form expression? $\endgroup$
    – user63495
    Commented Nov 22, 2014 at 13:58
  • $\begingroup$ @Camil: "The non-closed-formedness of that answer is that the size of the given expression depends on the size of the problem.", so i think it means, although you can find various method to simplify the present method, to eliminate the dots, but when you calculate it, it always need to expand it, and then it's still depends on the size, so it's still a non-closed-formed. Because the simplify of the writing method won't reduce the calculation work. It's just a method to simplify the writhing or remembering. $\endgroup$
    – diverger
    Commented Nov 22, 2014 at 14:58
  • $\begingroup$ @diverger yes, I get that (I also added an answer to the question on EE claiming that it's not possible to find a non-recursive formula 'without dots' for this), but now I'm just wondering in general about other finite continued fractions :) $\endgroup$
    – user63495
    Commented Nov 22, 2014 at 15:00
  • $\begingroup$ :), cheers, i think we can take a rest now. $\endgroup$
    – diverger
    Commented Nov 22, 2014 at 15:01
  • $\begingroup$ "But a finite continued fraction that is defined 'without dots' [] would be a closed-form expression?" Yes. $\endgroup$
    – user21467
    Commented Nov 22, 2014 at 19:41

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