= Solution
For fixed prior parameters, divide numerator and denominator by $n$:
$$
\mathbb E[p\mid x]=\frac{1+a/n}{S/n+(a+b)/n}\longrightarrow\frac1{\mathbb E X}=p_0,
$$
under independent sampling from a <geometric distribution> with true success probability $p_0>0$. The <law of large numbers> gives $S/n\to1/p_0$. In a large finite sample the <posterior mean> is close to $n/S=1/\bar x$, the <maximum-likelihood estimate>. The fixed prior's influence vanishes, and the estimate is <consistent>.
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