Convergence of a serie to a value

Question

I have to prove that $\sum_{n=0}^\infty a_n z^n = \frac{1-\theta}{1-\theta z}$ for some $\theta\in (0,1]$, $z\in (0,1]$.

I know that $a_n \geq 0$ and $\sum_{n=0}^\infty a_n=1$.

But hw can I choose $\theta$ ? And how can I get this equality ?

Answer 1

I assume that $\theta$ depends on $z$. Given $z\in(0,1]$, let $A=\sum_{n=0}^\infty a_nz^n$. Then $$ A=\frac{1-\theta}{1-\theta\,z}\implies \theta=\frac{1-A}{1-A\,z}. $$ Since $A,\theta\in(0,1]$, it follows that also $\theta\in(0,1]$.