= Factorial-gap power series
{title2=$\sum_{n=0}^\infty z^{n!}$}
This <lacunary power series> has <radius of convergence> one. For $|z|<1$, the inequality $n!\geq n$ for $n\geq1$ 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.
Back to article page
