The function $w(z)$ written implicitly as $\displaystyle f_n(z,w)=z-\prod_{j=1}^{j=n} (w-j)^j=0,\quad n\geq 2$ has $j$ finite singular points and ramifies into $1,2,\cdots,n-$cycle branches at the origin. For $n=2$ through $n=30$, the $1$-cycle branch has a radius of convergence extending to the most distant singular point. Is this always the case or is there an $n$ for which this does not hold?
One method to study the radius of convergence is by extrapolating an adequate number of expansion terms via the Root Test. For example, the following Mathematica code is used to generate $25$ terms of the $1$-cycle expansion of $f_5$. Plotting the absolute value of the $5$ singular points along the vertical axis (the blue points), and the Root Test results as the accompanying curve, it's clear which singular point the Root Test for this branch is converging to. This behavior is exhibited for $n=2,3,\cdots,30$.
theFunction = Expand[z - Product[(w - j)^j, {j, 1, 5}]];
aSeries = w /. AsymptoticSolve[theFunction == 0, {w, 1}, {z, 0, 25}];
