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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 $\sum_{n,m \geq 0} a_{mn} x^{\max\{m, \ n\}}$ ?

MAS
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  • What mode of convergence are you working with for the double series? There are many different ways to define the convergence, such as $$\lim_{F\uparrow\mathbb{N}0^2}\sum{(m,n)\in F}a_{mn},\qquad\lim_{M,N\to\infty}\sum_{n\leq N}\sum_{m\leq M}a_{mn}, \qquad \lim_{N\to\infty}\sum_{n\leq N}\left(\lim_{M\to\infty}\sum_{m\leq M}a_{mn}\right),\qquad\cdots$$ – Sangchul Lee Feb 27 '19 at 10:31
  • @SangchulLee, Convergent or divergence in $\mathbb{R}$ . How to decide whether the power series $ \sum {m,n \geq 0} a{mn} x^{n} x^m$ converges in $\mathbb{R}$ or not ? – MAS Feb 27 '19 at 11:16
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    I mean, there are different ways of defining what it means by a (double) series. Even in single-series case, we may define $\sum_n a_n$ either as the usual sense, i.e., the limit of the sequence of partial sums $\sum_{n\leq N}a_n$ as $N\to\infty$, or as the limit of the net of finite sums $\sum_{n\in F}a_n$ indexed by finite sets $F\subset\mathbb{N}_0$ as $F \uparrow \mathbb{N}_0$, which essentially corresponds to absolute convergence (a.k.a. $\ell^1$-convergence). In double-series case, situation is much more complicated. – Sangchul Lee Feb 27 '19 at 12:28
  • @SangchulLee, I mean the same concept of convergence $ \sum a_n x^n$ in $\mathbb{R}$ – MAS Feb 27 '19 at 15:26

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