I get counter example/proof for all other possibilities. But this one I couldn't do.
2026-03-28 05:23:11.1774675391
Is it possible to get positive radius of convergence for composition of two formal power series, if none of them has positive RoC.
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Take any power series $F=a_1 X+a_2 X^2+\ldots$ with $a_1\not=0$ with radius of convergence $0$ and $G:=F^{-1}$. Then $G$ also has radius of convergence $0$ because otherwise also $F=G^{-1}$ would have a positive radius of convergence. Finally $F\circ G=X$ has $\infty$ as its radius of convergence.