Let $X$ and $y$ be independent, identically distributed random variables with probabilities 

$$P(X=n)=\left\{\begin{matrix}
pq^{n-1}, n\geq 1\\ 
0, n<1
\end{matrix}\right.$$
$$P(Y=m)=\left\{\begin{matrix}
pq^{m-1}, m\geq 1\\ 
0, m<1
\end{matrix}\right.$$  

where $0<p<1$ and $p+q=1$. Evaluate the probabilities of the sum of $X$ and $Y$
$$P(Z=r)=P(X+Y=r).$$ 

**My solution so far:** 

Note that $P(X=n)=f\_{X}(n)$ and $P(X=m)=f\_{Y}(m)$.

$$P(Z=r) = P(Z=X+Y) = f\_{Z}(z).$$
$$f\_{Z}(z) = \sum\_{{\left \{ (n,m)|n+m=r \right \}}}P(X=n,Y=m)$$
$$=\sum\_{n}P(X=n,Y=z-n)=\sum\_{n}f\_{X}(n)f\_{Y}(z-n)$$
$$=\sum\_{n}pq^{n-1}pq^{z-n-1}= p^{2}q^{z-2}\sum\_{n}q^{0}= p^{2}q^{z-2}\sum\_{n=1}^{r-1}1 = p^{2}q^{z-2}(r-1).$$

EDIT - I believe I got it now.

It seems that the sum diverges to infinity, which is not correct. Is there anything wrong here?

Thanks for answering!!