We consider the Markov chain $\left(X_{n}\right)_{n\geq0}$ on $\mathbb{N}$ whose transition matrix $P=\left(p_{k,l}\right)_{k,l\geq0}$ given by
$$\forall k\in\mathbb{N} p_{k,0}=q_{k},p_{k,k+1}=p_{k}$$
with $p_{k}+q_{k}=1$ and $p_{k}>0$ for any $k\geq0$ .
Assume that $\left(X_{n}\right)_{n\geq0}$ is irreducible. Set $\tau=\textrm{inf}\left\{ n\geq1:X_{n}=0\right\}$ (with the convention $\textrm{inf}\textrm{Ø}=+\infty$ ); compute the value of $\mathbb{P}_{0}\left[\tau\geq n\right]$ and $\mathbb{P}_{0}\left[\tau=+\infty\right]$
Give me some hints please. Thank you in advance.