Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES AND LIMITSAgain using the Maclaurin expansion of exp x given in subsection 4.6.3, we notice thatS(θ) = Re [exp(exp iθ)] = Re [exp(cos θ + i sin θ)]=Re{[exp(cos θ)][exp(i sin θ)]} = [exp(cos θ)]Re [exp(i sin θ)]=[exp(cosθ)][cos(sin θ)]. ◭4.3 Convergence of infinite seriesAlthough the sums of some commonly occurring infinite series may be found,the sum of a general infinite series is usually difficult to calculate. Nevertheless,it is often useful to know whether the partial sum of such a series converges toa limit, even if the limit cannot be found explicitly. As mentioned at the end ofsection 4.1, if we allow N to tend to infinity, the partial sumN∑S N =u nn=1of a series may tend to a definite limit (i.e. the sum S of the series), or increaseor decrease without limit, or oscillate finitely or infinitely.To investigate the convergence of any given series, it is useful to have availablea number of tests and theorems of general applicability. We discuss them below;some we will merely state, since once they have been stated they become almostself-evident, but are no less useful for that.4.3.1 Absolute and conditional convergenceLet us first consider some general points concerning the convergence, or otherwise,of an infinite series. In general an infinite series ∑ u n can have complex terms,and in the special case of a real series the terms can be positive or negative. Fromany such series, however, we can always construct another series ∑ |u n | in whicheach term is simply the modulus of the corresponding term in the original series.Then each term in the new series will be a positive real number.If the series ∑ |u n | converges then ∑ u n also converges, and ∑ u n is said to beabsolutely convergent, i.e. the series formed by the absolute values is convergent.For an absolutely convergent series, the terms may be reordered without affectingthe convergence of the series. However, if ∑ |u n | diverges whilst ∑ u n convergesthen ∑ u n is said to be conditionally convergent. For a conditionally convergentseries, rearranging the order of the terms can affect the behaviour of the sumand, hence, whether the series converges or diverges. In fact, a theorem dueto Riemann shows that, by a suitable rearrangement, a conditionally convergentseries may be made to converge to any arbitrary limit, or to diverge, or to oscillatefinitely or infinitely! Of course, if the original series ∑ u n consists only of positivereal terms and converges then automatically it is absolutely convergent.124