I have the following problem
let $\{a_k\}_{k\geq0}$ be a sequence of positive numbers and $\{X_k\}$ a Markov chain with the probabilities $p_{k,0}=a_k$ and $p_{k,k+1}=1-a_k$, $k=0,1,2...$ Show that the chain is transient if, and only if, $\sum_ka_k<\infty$
This chain is irreducible, as every state communicate with which other. So, to prove it's transient I just need to show that any state is transient. So I picked state $0$. I also have the $j$ is transient if, and only if, $$\sum_{k=1}^{\infty}P_{jj}^k<\infty$$ If $P_{00}^k=a_k$ it'd be perfect. So I tried to get $P_{00}^1,P_{00}^2,P_{00}^3$, but the calculations are really ugly. Any help will be appreciated.