Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES AND LIMITSsome sort of relationship between successive terms. For example, if the nth termof a series is given byu n = 1 2 n ,for n =1, 2, 3,...,N then the sum of the first N terms will beS N =N∑n=1u n = 1 2 + 1 4 + 1 8 + ···+ 12 N . (4.1)It is clear that the sum of a finite number of terms is always finite, providedthat each term is itself finite. It is often of practical interest, however, to considerthe sum of a series with an infinite number of finite terms. The sum of aninfinite number of terms is best defined by first considering the partial sumof the first N terms, S N . If the value of the partial sum S N tends to a finitelimit, S, asN tends to infinity, then the series is said to converge and its sumis given by the limit S. In other words, the sum of an infinite series is givenbyS = limN→∞S N ,provided the limit exists. For complex infinite series, if S N approaches a limitS = X + iY as N →∞, this means that X N → X and Y N → Y separately, i.e.the real and imaginary parts of the series are each convergent series with sumsX and Y respectively.However, not all infinite series have finite sums. As N →∞, the value of thepartial sum S N may diverge: it may approach +∞ or −∞, or oscillate finitelyor infinitely. Moreover, for a series where each term depends on some variable,its convergence can depend on the value assumed by the variable. Whether aninfinite series converges, diverges or oscillates has important implications whendescribing physical systems. Methods for determining whether a series convergesare discussed in section 4.3.4.2 Summation of seriesIt is often necessary to find the sum of a finite series or a convergent infiniteseries. We now describe arithmetic, geometric and arithmetico-geometric series,which are particularly common and for which the sums are easily found. Othermethods that can sometimes be used to sum more complicated series are discussedbelow.116