# Is a finite continued fraction a closed-form expression?

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?

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.