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\section{Closed Forms from Mathematica}  Mathematica seems to offer closed forms for some of the expressions above. For example  \begin{equation}  K_{k=1}^\infty \frac{\left[\underset{k=1}{\overset{\infty}{\large K \normalsize}}  \frac{1}{x} = \frac{2}{x+\sqrt{4+x^2}} \frac{2}{x+\sqrt{4+x^2}}\\  \frac{\left[\underset{k=1}{\overset{\infty}{\large K \normalsize}} \frac{n}{x} = \frac{2n}{x+\sqrt{4n+x^2}}\\  \end{equation}  where we then see that the Laurent expansion does indeed follow the form \begin{equation}  \lim_{x\to\infty}\frac{2}{x+\sqrt{4+x^2}}=\sum_{k=1}^\infty \frac{C(k)}{x^{2k+1}}  \end{equation} with $C(k)$ the Catalan numbers.