# Finding power series with radius of convergence $r$

I'm working on this problem in real analysis:

Problem: Find for every $r \in \mathbb{R}^+ \setminus \left\{0 \right\}$ a power series $\sum_{n \to \infty} c_n z_n$ whose radius of convergence is $r$.

Attempt: In general, if we have a power series $$\sum_{n \to \infty} c_n (z - a)^n$$ centered around $a$, the radius of convergence equals $$\lim_{n \to \infty} \frac{ | c_n |}{ | c_{n+1} | }$$ provided that this limit exists. So I was looking for coefficients so that the limit of the ratio would give me $r$. But I couldn't find any, and I'm not sure if this is the correct method to approach this.

Assuming $r>0$. One may just take $$\sum_{n\geq0}^\infty\frac{z^n}{r^n}, \quad |z|<r.$$