= Solution
The first return time $\tau_1$ has distribution $\mathbb P_1(\tau_1=k)=\alpha_k$. It is finite almost surely, so state one is recurrent. It is <positive recurrent> precisely when its mean return time is finite:
$$
\boxed{\mathbb E_1\tau_1=\sum_{k\geq1}k\alpha_k<\infty.}
$$
By irreducibility the same recurrence class contains every state. Thus the condition is finiteness of the <expected value> of the positive-integer distribution $(\alpha_k)$. If that mean is infinite, the chain is null recurrent, not transient.
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