Let $\{X_n\}$ be a time-homogeneous Markov chain on $\mathbb{N}$ with $P(X_{2}=2 \,|\, X_1 = 1) = 1$, $P(X_2 = 1 \,|\, X_1 = n) = \frac{1}{n^\alpha}$, and $P(X_2 = n+1 \,|\, X_1 = n) = 1-\frac{1}{n^\alpha}$ for $n \geq 2$.
How can show that all states are transient when $\alpha > 1$ and all states are recurrent when $\alpha = 1$? I would think maybe you need an additional requirement that maybe we need irreducible chain but maybe this is not needed. I am told to conisder $\tau = \inf\{n:X_n=1\}$, but do not know how to use $\tau$.
Edit: I can show $P_n(\tau = k) = f_k(n,1)$ with $f_k(n,1) = P(X_1 \neq 1, \dots, X_{k-1} \neq 1,X_k=1 \,|\, X_0 = n)$. This can show that $$\sum_{k=1}^\infty kf_k(n,1) = \sum_{k=1}^\infty P_n(\tau \geq k).$$
Let $\tau_k^+:=\inf\{n \ge 1: X_n=k\}.$ Then for each possible initial state $k \ge 2$, $$P_k(\tau_k^+ =\infty) = \prod_{m \ge k} (1-m^{-\alpha}) \,. \tag{*}$$ This infinite product is strictly positive if $\alpha>1$ since $\sum_m m^{-\alpha}$ converges [1]. Thus all states $k \ge 2$ are transient, and the conclusion follows for $k=1$ as well.
The infinite product in $(*)$ tends to 0 for $\alpha=1$ since the harmonic series diverges. This implies recurrence.
[1] https://en.wikipedia.org/wiki/Infinite_product#Convergence_criteria