If I have a function $f(z)$ which is complex analytic on some neighbourhood of zero I know I can find a representation of the inverse as a series, and this series converges on some neighbourhood of $f(0)$. This is my understanding of the Lagrange inversion theorem. However, the formula for the coefficients of the series is not computationally friendly, expressed in terms of $n$-th derivatives and $n$-th powers of the variable $z$. In the same Wikipedia entry, a formula is given for formal power series which has an explicit formula for the coefficients of the inverse expressed in terms of the Bell polynomials. Specifically, if $f_n$ and $g_n$ are the $n$-th coefficient of Taylor series of $f(z)$ and the inverse series respectively, and $f_0 = 0$, $f_1 \neq 0$, then $$ g_n = \frac{1}{f^n_1}\sum_{k=1}^{n-1}(-1)^{k}n^{(k)}B_{n-1,k}(\hat{f}_1,\hat{f}_2,\ldots,\hat{f}_{n-k}),\quad n\geq 2, $$ where, $$ \hat{f}_k = \frac{f_{k+1}}{(k+1)f_1},\quad g_1 = \frac{1}{f_1} ,\quad \text{and } \quad n^{(k)}=n(n+1)\ldots(n+k-1),$$
and $B_{n-1,k}$ is the $(n-1,k)$-th partial exponential Bell polynomial. A similar formula, I think the same one actually although I haven't verified this, is used by Mathematica to invert series (see for example Series Reversion).
My question is: When can I assume this formal series actually represents the the inverse function? For me it would be enough to know that the formal series converges in some neighbourhood to the function, I don't need details about a specific radius of convergence.
