Let $X$ be the set of $\mathbb{R}$-valued sequences, i.e. $X := \mathbb{R}^{\mathbb{N}}=\{f: \mathbb{N} \to \mathbb{R}\}$, and let $S$ the set of sequences which can be expressed in closed form, i.e.: $$ S:= \{f \in \mathbb{R}^\mathbb{N} \space | \space f \text{ is in closed form}\} \subseteq X $$ Now since "closed form" is not well-defined: I basically mean the usual stuff. That is: $f$ is in closed form if there is a mathematical expression that can be evaluated in a finite number of operations. The allowed symbols in the expression are: constants, variables and applications of $!$ (factorial), $\exp, \ln$, the trigonometric and hyperbolic functions with their inverses, $\lfloor \cdot \rfloor, \lceil \cdot \rceil, [\cdot]$.

For example, $(a_k)_{k\in\mathbb{N}}\in S$ if $\displaystyle a_k = e^{(k!)^2\cdot \sin\left(\binom{2k}{k}\right)}$.

Now we have $S \subset X$, i.e. there are sequences which cannot be represented in closed form.

However, does at least the following statement hold? $$ \forall f \in X: \exists g \in S: f \leq g $$ (i.e. every sequence is bounded by a sequence which can be expressed in closed form)

  • $\begingroup$ How about $a_1=3$, $a_{n+1}=a_n!$ ? $\endgroup$ – Mindlack Jan 9 '19 at 18:29
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    $\begingroup$ Since you admit [arbitrary real] constants in the expressions deemed "closed-form", and there are continuum-many reals, there are at least so many sequences in $S$ (not that this makes difficulty in providing an answer). $\endgroup$ – John Bentin Jan 9 '19 at 18:54
  • $\begingroup$ @JohnBentin oh...you are absolutely right. I'll correct that. $\endgroup$ – Jakob B. Jan 9 '19 at 19:16

I think your definition of closed form is equivalent to the primitive recursive functions. The Ackermann function eventually overtakes every primitive recursive function, so the answer is that no function in $S$ dominates it.

In particular, probably the fastest growing type of function in your library is $n^{n^{n^n}}$ for some finite height of the tower. Ackermann uses $2$'s instead of $n$'s, but makes the height increase without bound, so eventually it will be taller than whatever tower you pick.

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I think not. Since $S$ is countable, enumerate it as $\{s(n)\}$. Now create a sequence $t$ such that

$$ t(n)_n = 1 + \max\{ s(n)_k \ | \ k \le n \} . $$.

This construction should work even if you allow constructions like repeated factorials $!^{n}$ as in @Mindlack 's comment.

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  • $\begingroup$ @SmileyCraft Any number that's not the $k$th term in $s(n)$. $\endgroup$ – Ethan Bolker Jan 9 '19 at 18:39
  • $\begingroup$ You mean $t_n > s(n)_n$, not just unequal (and there's diagonalization in there too)? $\endgroup$ – Daniel Schepler Jan 9 '19 at 18:39
  • $\begingroup$ I think it is useful to give $t$ explicitly, for example by $t(n)_k:=\max\{s(n)_l:1\leq l\leq k\}+1$. $\endgroup$ – SmileyCraft Jan 9 '19 at 18:40
  • $\begingroup$ @SmileyCraft Thanks. Used your suggestion. $\endgroup$ – Ethan Bolker Jan 9 '19 at 18:44
  • $\begingroup$ Interestingly this also proves that you can not define a bijection from $\mathbb{N}$ to $S$. $\endgroup$ – SmileyCraft Jan 9 '19 at 18:47

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