ABSTRACT
Since a series is a sum with an infinite number of terms (Section 21.1), it can be (i) convergent, (ii) divergent, or (iii) oscillating depending on whether its sum is, respectively (i) finite and unique, (ii) infinite, or (iii) takes more than one finite value. The sum with a finite number of terms has the association (commutation) properties, and these extend to convergent (Section 21.1) [absolutely convergent (Section 21.3)] series, whose terms may be associated (Section 21.2) [deranged (Section 21.1)] without changing the sum; a series is absolutely convergent (Section 21.3) if the series of moduli converges, this being a stricter requirement than simple convergence. The sum (Section 21.1) [product (Section 21.4)] of a series that involve association (rearrangement) of terms, applies both to simply and absolutely convergent series. Another concept of convergence, stronger than simple, and distinct from absolute, is uniform convergence (Section 21.5); the limit, derivate, and integral operators can be applied term-by-term (Section 21.6) to a uniformly convergent series. The strongest concept of convergence, is the totally convergent series (Section 21.7) that is, both, absolutely and uniformly convergent so that all relations and operations indicated above, and listed in Table 21.1, apply. A series of complex functions, for example, a geometric series (Section 21.8), may have different types of behavior, viz., divergence, oscillation, and simple, absolute, uniform, or total convergence, at distinct points, curves, or regions of the complex plane (Section 21.9). The power series can be seen as the successive approximations of a function by polynomials of increasing degree to obtain an arbitrary small error, provided that the series be convergent. For many problems that have no solution in finite form, the series provides a convenient approach, provided that its convergence properties allow the required operations.
