Find the radius of convergence of
\(\sum a^k z^k\) for \(a\) a constant.
\(\sum a^{k^2}z^k\)
Solution
By Cauchy–Hadamard, for \(\sum c_kz^k\), \[ {1\over R}=\limsup_{k\to\infty}|c_k|^{1/k}. \]
For \(c_k=a^k\), \[ {1\over R}=|a|, \] so \[ R= \begin{cases} \infty,&a=0,\\ |a|^{-1},&a\neq0. \end{cases} \]
For \(c_k=a^{k^2}\), \[ {1\over R} =\limsup_{k\to\infty}|a|^k. \] Hence \[ R= \begin{cases} \infty,&|a|<1,\\ 1,&|a|=1,\\ 0,&|a|>1. \end{cases} \]