I created a function $\eta(x)$ that was defined as $$\eta(x)=\sum_{n=0}^\infty \frac{1}{\operatorname{ack}(x,2,n)}$$ Where $\operatorname{ack}(x,y,z)$ is the original Ackermann function. From what I can tell, $\eta(1)$ is not defined as $\operatorname{ack}(1,x,n)=1$ for $n\geq2$. Also $\eta(2)$ will not converge, because $\frac{1}{\operatorname{ack}(2,2,n)}=\frac{1}{4}$ for any $n$. $\eta(3)$ does converge and so do other $\eta(n)$ for $n\geq3$. Does there exist a closed form solution for any $\eta(n)$? Are there even closed form solutions for $\eta(3)$ or $\eta(4)$?
Note: There is no particular reason I chose eta other than the fact that eta seemed to be an under-used letter in pure mathematics if it is purely reserved for something else I can change it.
Edit: The definition of the Ackermann function that I'm using is Ackermann's original definition:
$\operatorname {ack}(a,b,0)=a+b$
$\operatorname {ack}(a,0,1)=0$
$\operatorname {ack}(a,0,2)=1$
$\operatorname {ack}(a,0,c)=a$ where $c>2$
$\operatorname {ack}(a,b,c)=\operatorname {ack}(a,\operatorname {ack}(a,b-1,c),c-1)$ for any $a>0$ and $c>0$