Gompertz constant

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The Gompertz constant can be written as $$G=-e\mathrm{Ei}(-1)=\frac{1}{2-\frac{1}{4-\frac{4}{6-\frac{9}{8-\frac{16}{10-\frac{25}{12-\frac{36}{14-\frac{49}{16-\dots}}}}}}}},$$ where $e$ denotes the base of the exponential function and $\mathrm{Ei}$ denotes the exponential integral Ei.