Mathematical Methods for Physics and Engineering - Matematica.NET

SERIES AND LIMITSalthough in principle infinitely long, in practice may be simplified if x happens tohave a value small compared with unity. To see this note that P (x) forx =0.1has the following values: 1, if just one term is taken into account; 1.1, for twoterms; 1.11, for three terms; 1.111, for four terms, etc. If the quantity that itrepresents can only be measured with an accuracy of two decimal places, then allbut the first three terms may be ignored, i.e. when x =0.1 orlessP (x) =1+x + x 2 +O(x 3 ) ≈ 1+x + x 2 .This sort of approximation is often used to simplify equations into manageableforms. It may seem imprecise at first but is perfectly acceptable insofar as itmatches the experimental accuracy that can be achieved.The symbols O and ≈ used above need some further explanation. They areused to compare the behaviour of two functions when a variable upon whichboth functions depend tends to a particular limit, usually zero or infinity (andobvious from the context). For two functions f(x) andg(x), with g positive, theformal definitions of the above symbols are as follows:(i) If there exists a constant k such that |f| ≤kg as the limit is approachedthen f =O(g).(ii) If as the limit of x is approached f/g tends to a limit l, wherel ≠0,thenf ≈ lg. The statement f ≈ g means that the ratio of the two sides tendsto unity.4.5.1 Convergence of power seriesThe convergence or otherwise of power series is a crucial consideration in practicalterms. For example, if we are to use a power series as an approximation, it isclearly important that it tends to the precise answer as more and more terms ofthe approximation are taken. Consider the general power seriesP (x) =a 0 + a 1 x + a 2 x 2 + ··· .Using d’Alembert’s ratio test (see subsection 4.3.2), we see that P (x) convergesabsolutely if∣ ∣ ∣∣∣ a n+1ρ = lim xn→∞ a n∣ = |x| lima n+1 ∣∣∣n→∞ ∣ < 1.a nThus the convergence of P (x) depends upon the value of x, i.e. there is, in general,a range of values of x for which P (x) converges, an interval of convergence. Notethat at the limits of this range ρ = 1, and so the series may converge or diverge.The convergence of the series at the end-points may be determined by substitutingthese values of x into the power series P (x) and testing the resulting series usingany applicable method (discussed in section 4.3).132