Infinite Series

An infinite series asks you to add infinitely many numbers, and the hard part is never the addition — it is whether the sum exists at all. This course builds the idea in four layers. It starts with the definition of a series and of convergence, and with the relation between a partial sum and the limit of a sequence. It then introduces the model series you have to know — geometric, p-series, harmonic, and their logarithmic and exponential relatives — because they serve both as the reference points for convergence tests and as the prototypes of Taylor expansion. Next come the convergence tests themselves, together with guidance on which one to reach for. The course closes with power series and the Taylor expansion of a function, and what they are used for in approximation.