I have a power series $\displaystyle\sum_{m,n\geq0}c_{mn}x^my^n$, and I know that it converges absolutely on the upper triangle $\mathcal{T}=\{(x,y):0\leq x\leq y\leq1\}$.
I know that the domain of absolute convergence of a power series in several (real or complex) is a logarithmically convex Reinhardt domain but I am having difficulty seeing the application of that fact in this context.
My question is: What region in $\mathbb{R}^2$ is guaranteed to be contained in the domain of absolute convergence for the series, given that that domain contains the triangle $\mathcal{T}$?
Hint: Consider
$$\sum_{n=1}^{\infty}\left (\frac{x^n}{n^2} + \frac{y^n}{n^2}\right ).$$