SEQUENCES AND SERIES

Sequences and series have found a wide range of applications in both pure and applied mathematics. This chapter introduces the basic concepts of sequences and series and discusses their most commonly used properties. It presents some of the most commonly used tests, which are also sufficient for absolutely convergent series or series with positive terms. For series with positive terms, the absolute value signs are superfluous. For absolutely convergent series or series with positive terms, grouping – that is, inserting parentheses – yields a new convergent series having the same sum as the original series. A commonly used test for uniform convergence is the Weierstrass M-test or in short, the M-test. Uniform convergence is very important in dealing with series of functions. The chapter provides the most important and frequently used properties of uniformly convergent series.