Suppose we have a double series $\sum_{n \geq 0} \sum_{m \geq 0} a_{mn}$ which converges.
How to consider the double power series?
I know $\sum_{n,m \geq 0} a_{mn} x^n x^m$ is a double power series about the center $(0,0)$.
Can I consider $\sum_{n,m \geq 0} a_{mn} x^{\max\{m, \ n\}}$ as a double power series?
Next,
How to conclude about the convergence of the double power series $\sum_{n,m \geq 0} a_{mn} x^n x^m$ and $\sum_{n,m \geq 0} a_{mn} x^{\max\{m, \ n\}}$ ?