The sequence $\{p_n\}$ is defined as follows:
$$p_1=\frac{b}{a},$$
$$p_n=\frac{b-c\sum\limits_{i=1}^{n-1}p_i}{na}, \qquad (n\geq 2).$$
where a, b and c are parameters satisfying $a\gt b\gt 1\gt c$.
Is $\sum\limits_{n=1}^{\infty}p_n$ convergent or divergent? Does it depend on the values of a, b and c?
Edit
Alright, to answer the "close" requests, let me provide the context (at the risk of introducing mathematically unnecessary and irrelevant details).
This problem arises when I was considering some one dimensional pursuit-evade problem. Specifically, the pursuer and evader both have speed 1, and the pursuer lags $b$ behind in the beginning. However, at every distance $a$ they travel, the pursuer gets a boost that propels him a certain distance towards the evader. That distance is proportional to the current distance between them, and inversely proportional to the total distance he has traveled. So for example, at the $a$ milestone, the pursuer gets a boost of $c\frac{b}{a}$; at the $2a$ milestone, he gets a boost of $c\frac{b-\frac{bc}{a}}{2a}$, etc. I was considering small $c$ and large $a$, wondering if the pursuer can ever catch up, or if the evader can keep him at bay for large $b$ (which means the summation of $p_n$ converges).
I tried to simplify $p_n$, but has made no progress so far.