10.8.1 Series with complex terms

Problem: Determine the circle of convergence of the given series. Find out if the series converges at the given point \( z_{1}, z_{2}, z_{3} \) (if yes, then how: absolutely or conditionally). Make the drawing. \begin{tabular}{|c|c|c|c|} \hline Series & \( z_{1} \) & \( z_{2} \) & \( z_{3} \) \\ \hline\( \sum_{n=1}^{\infty} \frac{(z+1-2 i)^{n}}{2^{n}(n+1) \ln ^{2}(n+1)} \) & 0 & \( 1+2 i \) & -1 \\ \hline \end{tabular}