Sequences and series have found a wide range of applications in both pure and applied mathematics. A sequence is a succession of numbers or functions that may or may not converge to a limit value. A natural way to generate sequences is to use partial sums of series. In this regard, the convergence of a series is synonymous with the convergence of the corresponding sequence of partial sums. In this chapter, we introduce the basic concepts of sequences and series and discuss their most commonly used properties. We can name the evaluation of definite integrals, calculation of limits, and series approximation of functions to any desired level of accuracy as being among the most frequently used techniques with series. Series solutions of differential and integral equations and perturbative techniques are other important applications of series.
An infinite sequence is defined by assigning a number to each positive integer and then by writing them in a definite order:
A finite sequence has a finite number of elements . In ...