$G = (V, E)$ is a weighed graph with its vertex set $V = \{1, 2, 3, \cdots\}$. Every vertex $n$ has an edge with $n+1$ and $2n$. In other words, if $n\geq 2$, then $n$ has an edge with $n-1$; if $n$ is even, then $n$ has an edge with $\frac{n}{2}$. The edge between $n$ and $n+1$ has weight $1$, and the edge between $n$ and $2n$ has weight $n^{-\alpha}$, where $\alpha<1$. Prove that the random walk on this weighed graph is not recursive.
I tried to find out the "resistance" between $0$ and $\infty$, but I failed. Does this attempt work? Or does there exist any better idea for the proof of the conclusion above? Perhaps I need some help.