Here is a cool sum. Is it possible to derive this closed form?.

I apologize for the wacky sums that did not pan out. As I said, I only had around 10 decimals places. It was safe to assume that closed forms like that were too good to be true

They came from that Inverse Symbolic Calculator. I rarely use it, so I was just playing around.

Anyway, try this one if you like. I think it is more amenable.

$$\sum_{n=1}^{\infty}\cot^{-1}(n^{2})=\cot^{-1}\left(\frac{1+z}{1-z}\right)$$

where $z=\cot(\frac{\pi}{\sqrt{2}})\tanh(\frac{\pi}{\sqrt{2}})$