Half-plane of convergence of an exponential power series
= Half-plane of convergence of an exponential power series
If the ordinary power series $\sum_{n\geq0}b_nw^n$ has radius $R\in(0,\infty)$, then
$$
\sum_{n\geq0}b_ne^{nz}
$$
converges for $\operatorname{Re}z<\log R$ and diverges for $\operatorname{Re}z>\log R$. Behavior on the boundary depends on the coefficients.