 Solution

The equal rooted counts imply that the self-avoiding walk generating function is $Z_G(x)={\sum }_{n\ge 0}{\sigma }_n{x}^n$. The Cauchy-Hadamard theorem identifies the radius of a power series $\sum a_n{x}^n$ as the reciprocal of $\mathrm{limsup}|a_n{|}^{1/n}$. ConsequentlyEquivalently, 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 ${\sigma }_n{\mu }^{-n}\ge 1$, so this particular series diverges there as well.