Tobias Holck Colding continues MIT's 18.100B Real Analysis course with the eighth lecture in the series, picking up the set of tests used to determine whether an infinite series converges. He introduces limsup and liminf, the tools needed to state these convergence tests in their most general form, working through the definitions and their role in distinguishing conditional behavior from strict bounds. The lecture then applies limsup to define the radius of convergence for a power series, treated as an infinite sum of polynomials. The session runs eighty-one minutes on the blackboard, building each definition from the previous one so the logic of why limsup is the right tool, rather than an ordinary limit, stays visible throughout. It assumes the prior seven lectures' groundwork in sequences and basic series tests.