The power series \(\sum_{n=1}^\infty nz^n\) has radius of convergence \(1\) and diverges at every \(z\) with \(\abs{z}=1\). Its radius is \(1\) because \(\limsup_n n^{1/n}=1\). For \(\abs{z}=1\) the terms satisfy \(\abs{nz^n}=n\not\to0\), so the series diverges.
Give a power series with radius of convergence $1$ that diverges at every point of $S^1$.