Let $X_1, X_2,... $ be a discrete renewal process, in which $X_i$ denotes the time in between renewals with distribution: $Pr(X_i=1)=p$ and $Pr(X_i=2)=q=1-p. $ I want to show that the renewal equation $$m(n)=E(N(n))=\frac{n}{1+q}-\frac{q^2}{(1+q)^2}[1-(-q)^n]$$
This is what I know: Let $$F_k(n)=Pr(T_k\le n)$$ Then $$m(n)= \sum_{k=1}^\infty F_k(n)$$ I've noticed that $Pr(T_k \le n)= 0$ for $ n\lt k$; $1-q^k$ for $k \le n \lt 2k$, and $1$, for $n \ge 2k$
which, btw im not 100% on. But now I dont know what to do with this since the summation diverges if I plug in $1-q^2$ for $F_k(n)$. (an goes to infinity). Any suggestions?
p.s. writing mathjax for the first time was a pain in the ass experience...