Mathematical Methods for Physics and Engineering : A Comprehensive Guide

4.5 POWER SERIES

Q(x)intoP(x)toobtainP(Q(x)), but we must be careful since the value ofQ(x)

may lie outside the region of convergence forP(x), with the consequence that the

resulting seriesP(Q(x)) does not converge.

(iii) If a power seriesP(x) converges for a particular range ofxthen the series

obtained by differentiating every term and the series obtained by integrating every

term also converge in this range.

This is easily seen for the power series

P(x)=a 0 +a 1 x+a 2 x^2 +···,

which converges if|x|<limn→∞|an/an+1|≡k. The series obtained by differenti-

atingP(x) with respect toxis given by

dP

dx

=a 1 +2a 2 x+3a 3 x^2 +···

and converges if

|x|<lim

n→∞

∣

∣

∣

∣

nan

(n+1)an+1

∣

∣

∣

∣=k.

Similarly the series obtained by integratingP(x) term by term,
∫
P(x)dx=a 0 x+

a 1 x^2

2

+

a 2 x^3

3

+···,

converges if

|x|<lim

n→∞

∣

∣

∣

∣

(n+2)an

(n+1)an+1

∣

∣

∣

∣=k.

So, series resulting from differentiation or integration have the same interval of

convergence as the original series. However, even if the original series converges

at either end-point of the interval, it is not necessarily the case that the new series

will do so. The new series must be tested separately at the end-points in order

to determine whether it converges there. Note that although power series may be

integrated or differentiated without altering their interval of convergence, this is

not true for series in general.

It is also worth noting that differentiating or integrating a power series term

by term within its interval of convergence is equivalent to differentiating or

integrating the function it represents. For example, consider the power series

expansion of sinx,

sinx=x−

x^3

3!

+

x^5

5!

−

x^7

7!

+···, (4.14)

which converges for all values ofx. If we differentiate term by term, the series

becomes

1 −

x^2

2!

+

x^4

4!

−

x^6

6!

+···,

which is the series expansion of cosx,asweexpect.