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Proposition 1.1.8 in this answer, from "Enumerative Combinatorics" by Stanley, says

The infinite series $\sum_j F_j(x)$ converges if and only if $\lim_{j \to \infty} \deg(F_j(x)) = \infty$.

where each $F_j(x)$ is a formal power series.

I must be misunderstanding something fundamental because this makes no sense to me. Consider the sequence of formal power series defined as $F_j(x) = 1 + x + x^2 + \cdots$ for all $j$. Then the series "converges" to $\infty + \infty x + \infty x^2 + \cdots$, which means that it doesn't converge, but it still satisfies the condition $\lim_{j \to \infty} \deg(F_j(x)) = \infty$... right?

mortimer
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    Here, the degree of a formal series refers to the minimum exponent with non-zero coefficient, not the maximum one. – Sangchul Lee Sep 21 '23 at 02:09
  • @SangchulLee Oh.. that answers my question, thanks. – mortimer Sep 21 '23 at 02:48
  • Essentially, this is equivalent to the requirement that each degree appear in only finitely many of the $F_j(x)$'s so that the coefficient at each $x^n$ in $\sum_j F_j(x)$ is only a finite sum. – Alexander Burstein Sep 21 '23 at 17:50

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