(0:00) Plan for the lecture. (0:44) Initial example for series: expand f(z) = 5/(3+7z) as a Taylor series about z = 8 by using some algebra tricks and the formula for the sum of a geometric series. (12:32) Sequences of real-valued functions of a real variable and pointwise convergence (examples on Mathematica). (24:55) Definition of convergence of a (complex) series as the limit of a sequence of partial sums (when the limit exists). (30:40) Geometric series example (find the function and find the disk of convergence, (almost) confirming the disk of convergence with the Ratio Test). (38:20) Give an illustration of the maximum modulus principle on Mathematica, optimizing the modulus over a closed disk by analyzing the behavior along the boundary of the disk.
Based on "Fundamentals of Complex Analysis, with Applications to Engineering and Science", by E.B. Saff and A.D. Snider (3rd Edition). "Visual Complex Analysis", by Tristan Needham, is also referred to a lot. Mathematica is often used, especially to visualize complex analytic (conformal) mappings.