Here is an idea for you The sequence can be written as follows
This matrix can be diagonalized and solved explicitly
For the two eigenvalues I got
I hope this helps.
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.
... that is 'linear and homogeneous' and the solution of which is...
... where and are 'arbitrary constants' that depend from and , and the solution of the 'characteristic equation'...
Now for is and , so that in any case is...