Lacunary power series (${\sum }_j{c}_j{z}^{m_j}$)

A power series is lacunary when only a sparse sequence of powers has nonzero coefficients. For ${\sum }_j{c}_j{z}^{m_j}$ with strictly increasing integer exponents, the radius of convergence satisfies $R^{-1}={\mathrm{limsup}}_j|c_j{|}^{1/m_j}$ by the Cauchy-Hadamard theorem. The exponent $m_j$, rather than the term index $j$, controls the coefficient root.

This lacunary power series has radius of convergence one. For $|z|<1$, the inequality $n!\ge n$ for $n\ge 1$ gives absolute convergence by a geometric series comparison. For $|z|>1$ its terms do not tend to zero. At every point of the unit circle the terms have modulus one, so the term test for divergence proves divergence there too. The two initial terms both equal $z$, since $0!=1!=1$; this finite repetition does not affect any convergence conclusion.