Monotone coefficients tending to zero force convergence at every boundary point except $1$.Suggest a slogan
Let \((c_k)_{k\ge0}\) be a nonincreasing sequence of nonnegative real numbers with \(c_k\to0\), and let \(f(z)\coloneqq\sum_{k\ge0}c_kz^k\). Then the series \(f(z)\) converges for every \(z\) with \(\abs{z}=1\) and \(z\neq1\).
Let \(\abs{z}=1\) with \(z\neq1\). The partial sums of \(\sum_k z^k\) satisfy \(\abs{\sum_{k=0}^{N}z^k}=\abs{\frac{1-z^{N+1}}{1-z}}\le\frac{2}{\abs{1-z}}\) for every \(N\), so they are bounded. Since \((c_k)\) is nonincreasing with limit \(0\), Dirichlet's test applied to \(a_k=c_k\) and \(b_k=z^k\) shows that \(\sum_k c_kz^k\) converges.