Define the following partial convergents in terms of simple continued fractions:
$$a_{0} = \frac{1}{1}, \quad a_{1} = \frac{1}{1+ \frac{1}{1}}, \quad a_{2} = \frac{1}{1+ \frac{1}{{1+\frac{1}{1}}}}, \ \dots $$
Then the limit of these partial convergents goes to the inverse of the golden ratio: $$\lim_{n \to \infty} a_{n} = \phi^{-1} = \phi -1.$$
I wonder whether convergent infinite series involving such partial convergents have been studied and evaluated. Consider, for instance, the following series:
\begin{align*} S_{1} &= \sum_{n=1}^{\infty} \left( a_{n} - \phi^{-1} \right), \text{ and} \\ S_{2} &= \sum_{k=1}^{\infty} \frac{a_{k}}{2^{k}} \ . \end{align*}
Question: has the closed form evaluation of infinite series involving continued fraction partial convergents, like $S_{1}$ and $S_{2}$, been studied and described in the literature before?