Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

4.5 POWER SERIES


Determine the range of values ofxfor which the following power series converges:
P(x)=1+2x+4x^2 +8x^3 +···.

By using the interval-of-convergence method discussed above,


ρ= lim
n→∞


∣∣



2 n+1
2 n

x


∣∣


∣=|^2 x|,

and hence the power series will converge for|x|< 1 /2. Examining the end-points of the
interval separately, we find


P(1/2)=1+1+1+···,
P(− 1 /2) = 1−1+1−···.

ObviouslyP(1/2) diverges, whileP(− 1 /2) oscillates. ThereforeP(x) is not convergent at
either end-point of the region but is convergent for− 1 <x<1.


The convergence of power series may be extended to the case where the

parameterzis complex. For the power series


P(z)=a 0 +a 1 z+a 2 z^2 +···,

we find thatP(z) converges if


ρ= lim
n→∞





an+1
an

z




∣=|z|nlim→∞





an+1
an




∣<^1.

We therefore have a range in|z|for whichP(z) converges, i.e.P(z)converges


for values ofzlying within a circle in the Argand diagram (in this case centred


on the origin of the Argand diagram). The radius of the circle is called the


radius of convergence:ifzlies inside the circle, the series will converge whereas


ifzlies outside the circle, the series will diverge; if, though,zlies on the circle


then the convergence must be tested using another method. Clearly the radius of


convergenceRis given by 1/R= limn→∞|an+1/an|.


Determine the range of values ofzfor which the following complex power series converges:

P(z)=1−

z
2

+


z^2
4


z^3
8

+···.


We find thatρ=|z/ 2 |, which shows thatP(z)convergesfor|z|<2. Therefore the circle
of convergence in the Argand diagram is centredon the origin and has a radiusR=2.
On this circle we must test the convergence by substituting the value ofzintoP(z)and
considering the resulting series. On the circle of convergence we can writez=2expiθ.
Substituting this intoP(z), we obtain


P(z)=1−

2expiθ
2

+


4exp2iθ
4

−···


=1−expiθ+[expiθ]^2 −···,

which is a complex infinite geometric series with first terma= 1 and common ratio

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