The power series \(\sum_{n=1}^\infty \frac{z^n}{n}\) has radius of convergence \(1\), diverges at \(z=1\), and converges at every other point of \(S^1\). Its radius is \(1\) because \(\limsup_n(1/n)^{1/n}=1\). At \(z=1\) it is the harmonic series, which diverges. For \(\abs{z}=1\) with \(z\neq1\), the partial sums satisfy \(\abs{\sum_{n=1}^N z^n}=\abs{\frac{z(1-z^N)}{1-z}}\le\frac{2}{\abs{1-z}}\), and \(1/n\) decreases to \(0\), so the series converges by Dirichlet’s test.
Give a power series with radius of convergence $1$ that converges at every point of $S^1$ except $z=1$.