Determine the domain of definition of $\sigma$ and justify that $\sigma$ is continuous on it, where $\sigma(x) = \sum_{k=1}^{+\infty} \frac{x^k}{k^2}$.
Determine the domain of definition of $\sigma$ and justify that $\sigma$ is continuous on it, where $\sigma(x) = \sum_{k=1}^{+\infty} \frac{x^k}{k^2}$.