Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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).