The power series \(\sum_{n=1}^\infty \frac{z^n}{n^2}\) has radius of convergence \(1\) and converges at every \(z\) with \(\abs{z}=1\). Its radius is \(1\) because \(\limsup_n(1/n^2)^{1/n}=1\). For \(\abs{z}=1\), \(\abs{z^n/n^2}=1/n^2\) and \(\sum_n 1/n^2<\infty\), so the series converges absolutely.
Give a power series with radius of convergence $1$ that converges at every point of $S^1$.