Maybe the question is obvious, but I don't know the answer
Let $p(x_{1},...,x_{r})$ be a polynomial with nonnegative coefficients and consider $$ F_{n}(x) = \sum_{k=0}^{n}(p(x))^{k} $$ where $x = (x_{1},...,x_{r}) \in \mathbb{R}^{r}$. Suppose $F_{n}$ converges uniformly to some function $f$. Now the coefficient of the taylor expansion of $F_{n}$ around $0$ are nonnegative. Can I conclude that the coefficients of the taylor epansion of $f$ around 0 are also nonnegative?
EDIT: This was an answer to the question before it was edited.
No. Take for example $f : \mathbb{R} \to \mathbb{R}$ defined by $$ f(x) = \frac{1}{1+x^2}. $$ Then by the geometric series formula $$ f(x) = \frac{1}{1-(-x^2)} = \sum_{k=0}^\infty (-1)^kx^{2k}, $$ whenever $|x| < 1$. By the uniqueness of Taylor expansions, we know that this is the Taylor expansion of $f$ at $0$, but its coefficients are certainly not nonnegative.