$$\frac{1}{1-z+qp^kz^{k+1}}, \space (p+q = 1 \space\space p, q > 0, \space k : positive \space integer \ge 2)$$
To find a pole, other than 1/p,
I got to know the pole is between 1 and 1/q.
Then, by using a fixed point iteration, $z_{n+1} = 1+qp^kz^{k+1}_n $, starting $z_0 = 1$(lowerbound)
I have two difficulties with that.
Q1) How to find the pole (using big O notation)?
(too many binomial coeffs came out... I can't handle them)
Q2) How about the upperbound? Starting $z_0 = 1/q$, but I think only in $p<q$, Otherwise, it won't reach the fixed point. Is there any other starting point in upperbound to reach the fixed point, regardless of $p > q \space or \space p < q$