Does every convergent infinite series have a closed-form value?

Question

Does every convergent infinite series have a closed-form value?

I apologize if this question seems totally crazy to some of you. There's a ton of series that converge, but only a fraction of them have a closed-form value, so how can I even ask such a question? Isn't it obvious that the answer is no?

Well, not so fast. My intention with this question is more like "Does the possibility exist that every convergent infinite series have a closed-form value?" or have we proven that it's impossible? A "simple" way to prove this would be to find an infinite convergent series and then show that it would be impossible for it to have a closed-form value. But how would you do that?

Sure, there is an enormous amount of series that converge, for which there is not a known closed-form value, e.g. one of the series I asked about earlier, and one may use this as an argument for the answer no, but that's not enough for me.

I hope this provokes some discussion and that you don't think this questions is completely dumb.

Answer 1

Depends on what you understand by "closed form". If you mean "expressible in a finite number of (give a list of functions and constants)", absolutely not. For a few examples, the values of Riemann's zeta function:

$\begin{align*} \zeta(s) = \sum_{n \ge 1} n^{-s} \end{align*}$

are known only for even values of $s$, their values for odd values of $s$ are a complete mystery.

Simpler, even: It is easy to see that there the set of convergent series is non-denumerable, even if you restrict yourself to rational terms; but the set of finite formulas made up of algegraic numbers, a handful of "well-known" trascendentals, and a finite set of functions is denumerable. There must be series with no "closed form".

Answer 2

I think the definition of "closed-form value" should include that it is expressible as a finite combination of some known constants via some algebraic (or transcendental, but still elementary) formula. Therefore, a series that converges to a number that is not computable does not have a closed form value. Since most real numbers are not computable, most series do not converge to a closed form value.

A number that is not computable is one where there is absolutely no computer program that will output all of the digits of the number, even given infinite time. In fact, since there are only countably many computer programs, only countably many real numbers are computable, even though it is, by definition, difficult to produce even one that is not. Any number with a closed form expression would be computable, and we could compute it by applying the formula.