Recursive Sequence Convergence

and r and s are positive numbers with p + q = 1. for all . Prove that is convergent and find limits in terms of , p and q.

Okay I began by letting p = 1 - q so I could eliminate that variable. Then I began working it out up to but I couldn't find any pattern in terms of coefficients of the polynomial. It's a polynomial with respect to p. I am trying to get this to work out to something that I can put into summation notation, prove it is monotone, and then find an upper and/or lower bound but I am unsure how to go about putting it in summation notation - it's entirely possible this is not the way to do it but it seems most logical to me. The first term (in order of descending exponent on p) of the expansion is always . The last term is when n is even and when n is odd. These are the only real patterns I've been able to discern though. Any insight is much appreciated. Thanks.