I am surprised by the question. My remarks are too long for a comment. The sums are bounded if $\alpha \ne 1$, since for every $n \ge 1$, $$\Big|\sum_{k=1}^n \Re R_\alpha^k z \Big| = \Big|\Re \Big(\sum_{k=1}^n R_\alpha^k z \Big) \Big| = \Big| \Re \Big(\frac{\alpha-\alpha^{n+1}}{1-\alpha}z\Big) \Big| \le \Big|\frac{\alpha-\alpha^{n+1}}{1-\alpha}z\Big| \le \frac{2}{|1-\alpha|}.$$
Then, what is the definition of the logarithm of a negative number? Whatever the choice done for the imaginary part of the logarithm (i.e. the argument) the real part of the real part of the logarithm of the sums is bounded above by some constant, so you cannot find a limit $1/2$.