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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.

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

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