I believe that this identity isn't known $$-\sum_{n=1}^\infty\frac{G_n}{n}(\mu-1)^n=\gamma+\log(\mu-1),\tag{1}$$ where $G_n$ is the $n$th Gregory coefficient, $\gamma$ the Euler-Mascheroni constant and $\mu$ the so-called Soldner's constant. I've deduced it from a simple but elaborated argument that involves some of the identities related to Gregory coefficients from this Wikipedia. But I'm unsure if it it is possible to deduce easily from a different way, or is equivalent (the same) to some identity from the litearute.
Question. Is it known the identity $(1)$ from the literature? Then answer this question as a reference request, and I try to search and read such proposition. In other case do you know a simple way to deduce $(1)$? In this last case, provide hints to get $(1)$ (as was said I know the formulas involving the Gregory coefficients that shows the cited Wikipedia's article). Many thanks.
That is I would like to know if $(1)$ has mathematical content or if it is obvious.
You can find the definition of the Soldner's constant from this MathWorld.
Your formula comes from the combination of:
$\displaystyle li(2)=\int_{\mu}^2\frac{dt}{\ln t}$
$\displaystyle \frac{1}{\ln t} = \frac{1}{t-1} + \sum\limits_{n=1}^\infty G_n (t-1)^{n-1}$
$\displaystyle \sum\limits_{n=1}^\infty \frac{G_n}{n}=li(2)-\gamma$