Because of the close relation of infinite series and improper integrals to the special functions, it can be useful to first review some basic concepts of series and integrals. Infinite products, which are generally less well known, are introduced here mostly for the sake of completeness, but in some instances they are also useful.
Infinite series are important in almost all areas of pure and applied mathematics. In addition to numerous other uses, they are used to define certain functions and to calculate accurate numerical estimates of the values of these functions. In calculus the primary problem is deciding whether a given series converges or diverges. In practice, however, the more crucial problem may actually be summing the series. If a convergent series converges too slowly, the series may be worthless for computational purposes. On the other hand, the first few terms of a divergent series in some instances may give excellent results. Improper integrals and infinite products are used in much the same fashion as infinite series, and, in fact, their basic theory closely parallels that of infinite series.
In the application of mathematics frequently two or more limiting processes have to be performed successively. For example, we often find the derivative (or integral) of an infinite sum of functions by taking the sum of derivatives (or integrals) of the individual terms of the series. However, in many cases of interest, performing two limit operations in one order may yield an answer different from that obtained using the other order. That is, the order in which the limiting processes are carried out may be critical.
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