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\end{equation}  For a given sequence it is not entirely clear if convergence will be achieved. We may take a partial series. For $n=24$ we have $C=0.461538461538461538461538653254437869822485207100604251251706...$ $C_{24}=0.461538461538461538461538653254437869822485207100604251251706...$  which appears to repeat at first with a $6$ digit sequence $.461538$. $.461538$, repeating $4$ times.  FOr $n=40$ we have $C_{40}=0.461538461538461538461538461538461538461557633136094674556213...$ which repeats 6.5 times. If we assume convergence we could then argue that \begin{equation}  C_\infty = 0.\overline{461538} =\frac{6}{13}  \end{equation}