Example 4: The power series $$\sum _{i=1}^{\infty }a_{i}z^{i}{\text{ where }}a_{i}={\frac {(-1)^{n-1}}{2^{n}n}}{\text{ for }}n=\lfloor \log _{2}(i)\rfloor +1$$has radius of convergence 1 and converges uniformly on the entire boundary $|z| = 1$, but does not converge absolutely on the boundary.
