Title: 16_2 Infinite Series and convergence tests
Course: HELM Workbooks 2023-2024 > Section: HELM 16: Sequences and Series

We extend the concept of a finite series, met in Section 16.1, to the situation in which the number of terms increase without bound. We define what is meant by an infinite series being convergent by considering the partial sums of the series. As prime examples of infinite series we examine the harmonic and the alternating harmonic series and show that the former is divergent and the latter is convergent. We consider various tests for the convergence of series, in particular we introduce the ratio test which is a test applicable to series of positive terms. Finally we define the meaning of the terms absolute and conditional convergence.