= Solution
The equal rooted counts imply that the <self-avoiding walk generating function> is $Z_G(x)=\sum_{n\geq0}\sigma_nx^n$. The <Cauchy-Hadamard theorem> identifies the radius of a <power series> $\sum a_nx^n$ as the reciprocal of $\limsup|a_n|^{1/n}$. Consequently
$$
\boxed{R_G=\mu^{-1}}.
$$
Equivalently, the <root test> gives convergence when $x\mu<1$ and divergence when $x\mu>1$. At the positive boundary $x=\mu^{-1}$, the preceding infimum formula gives $\sigma_n\mu^{-n}\geq1$, so this particular series diverges there as well.
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