Chapter 3 Sec. 13: Addition and Multiplication of Series [Principles of Mathematical Analysis]
Welcome back to Chapter 3 of Principles of Mathematical Analysis. In this section we ask two natural questions about the arithmetic of series. First, if two series converge, can we add them term by term? The answer is yes, and the proof takes only a few lines. Second, can we multiply two convergent series? This turns out to be much more subtle. We first have to decide what the product of two series should even mean, and that leads to the Cauchy product. Then we will see a striking example where the product of two convergent series diverges, a theorem of Mertens that rescues the situation when one of the factors converges absolutely, and finally a theorem of Abel describing what happens when the product series converges at all. Let's get started.
Slides: https://github.com/petercerno/princip...