In a previous question of mine, I asked for the probability distribution of an agent taking a biased walk on the positive integers (with a reflecting boundary at the origin):
Probability distribution for the position of a biased random walker on the positive integers
Here, for a $+1$ step for the walker has probability $p$, for a $-1$ step (away from the origin) the walker has probability $q$, and since $(p+q) \leq 1$, we allow the walker to stay in place with probability $r = 1 - (p+q)$. In order to prevent this walk from becoming transient, we set $p \leq q$.
By solving the relation:
$P(X=n)= p P(X=n-1) +(1-p-q) P(X=n) + q P(X=n+1)$
The user Henry provided a rather nice solution for the stationary distribution that sums to unity and matches up will with simulations with ~$10^7$ or more steps:
$P(X=n) = \left(\frac{p}{q}\right)^n \left(1-\frac{p}{q}\right)$
My question is as follows: Say we initialize our walking agent at some position $0 \leq i < M$, and we're interested in the average hitting time at position $M$. Say we also calculate some sum:
$S = \sum_{n=M}^{\infty} \left(\frac{p}{q}\right)^n \left(1-\frac{p}{q}\right)$
Considering that the stationary distribution $P(X=n)$ is a smoothly decreasing analytic function, can we use $S$ to say anything about the average hitting time, $H(M)$ for the walker agent at position $M$?