Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

SERIES AND LIMITS


although in principle infinitely long, in practice may be simplified ifxhappens to


have a value small compared with unity. To see this note thatP(x)forx=0. 1


has the following values: 1, if just one term is taken into account; 1.1, for two


terms; 1.11, for three terms; 1.111, for four terms, etc. If the quantity that it


represents can only be measured with an accuracy of two decimal places, then all


but the first three terms may be ignored, i.e. whenx=0.1orless


P(x)=1+x+x^2 +O(x^3 )≈1+x+x^2.

This sort of approximation is often used to simplify equations into manageable


forms. It may seem imprecise at first but is perfectly acceptable insofar as it


matches the experimental accuracy that can be achieved.


The symbols O and≈used above need some further explanation. They are

used to compare the behaviour of two functions when a variable upon which


both functions depend tends to a particular limit, usually zero or infinity (and


obvious from the context). For two functionsf(x)andg(x), withgpositive, the


formaldefinitionsof the above symbols are as follows:


(i) If there exists a constantksuch that|f|≤kgas the limit is approached
thenf=O(g).
(ii) If as the limit ofxis approachedf/gtends to a limitl,wherel=0,then
f≈lg. The statementf≈gmeans that the ratio of the two sides tends
to unity.

4.5.1 Convergence of power series

The convergence or otherwise of power series is a crucial consideration in practical


terms. For example, if we are to use a power series as an approximation, it is


clearly important that it tends to the precise answer as more and more terms of


the approximation are taken. Consider the general power series


P(x)=a 0 +a 1 x+a 2 x^2 +···.

Using d’Alembert’s ratio test (see subsection 4.3.2), we see thatP(x)converges


absolutely if


ρ= lim
n→∞





an+1
an

x




∣=|x|nlim→∞





an+1
an




∣<^1.

Thus the convergence ofP(x) depends upon the value ofx, i.e. there is, in general,


a range of values ofxfor whichP(x) converges, aninterval of convergence.Note


that at the limits of this rangeρ= 1, and so the series may converge or diverge.


The convergence of the series at the end-points may be determined by substituting


these values ofxinto the power seriesP(x) and testing the resulting series using


any applicable method (discussed in section 4.3).

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