> [!Example] Example (Geometric series I) > The series has $\sum x^{n}$ has a radius of convergence $R=1$. > [!Example] Example (Geometric series II) > The series has $\sum p^{n}x^{n},p \in \mathbb{R}$ has a radius of convergence $R=1/|p|$. > > **Proof** > The series converges iff $|px|<1\implies |x|<1/|p|$.