Lacunary power series
= Lacunary power series
{title2=$\sum_jc_jz^{m_j}$}
A power series is lacunary when only a sparse sequence of powers has nonzero coefficients. For $\sum_jc_jz^{m_j}$ with strictly increasing integer exponents, the <radius of convergence> satisfies $R^{-1}=\limsup_j|c_j|^{1/m_j}$ by the <Cauchy-Hadamard theorem>. The exponent $m_j$, rather than the term index $j$, controls the coefficient root.