I am on chapter 6 of "Understanding digital signal processing", 3rd Edition by Richard G. Lyons. But I am stuck on question 6.15. Here is the network in question.
I wrote down the difference equation as $$y(n)=x(n)+y(n-1)-Qy(n-1),$$ where $x(n)=D$(i.e. constant for all $n$), $Q$ is positive and less than $1$ and network is at rest at $n=0$ (i.e $y(n-1)=0$ when $n=0.$) In order to answer the question, I need to develop the series equation for $y(n)$ and find its closed form.
I have developed the series expression for $$y(n)=D\sum^n_{k=0}(-Q)^k {{(n+1)}!\over{(k+1)!(n-k)!}}.$$
And the terms without simplifying the factorials are $$y(n)=D{(n+1)!\over n!} - DQ{(n+1)!\over 2(n-1)!} +DQ^2{(n+1)!\over 6(n-2)!} -DQ^3{(n+1)!\over 24(n-3)!} . .. +(-Q)^nD$$.
My issue is that I cannot find where I have gone wrong because this series for $y(n)$ does not seem to have a common ratio which makes it hard for me to write it in the closed form as in $$\sum^{N-1}_{n=p}r^n={r^p-r^N\over{1-r}}.$$
My question is, is my developed $y(n)$ suitable? If it is suitable, does it have a closed form?
Thanks