Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES AND LIMITS

x=a+hin the above expression. It then reads

f(x)=f(a)+(x−a)f′(a)+

(x−a)^2

2!

f′′(a)+···+

(x−a)n−^1

(n−1)!

f(n−1)(a)+Rn(x),

(4.18)

where the remainder now takes the form

Rn(x)=

(x−a)n

n!

f(n)(ξ),

andξlies in the range [a, x]. Each of the formulae (4.17), (4.18) gives us the

Taylor expansionof the function about the pointx=a. A special case occurs

whena= 0. Such Taylor expansions, aboutx= 0, are calledMaclaurin series.

Taylor’s theorem is also valid without significant modification for functions

of a complex variable (see chapter 24). The extension of Taylor’s theorem to

functions of more than one variable is given in chapter 5.

For a function to be expressible as an infinite power series we require it to be

infinitely differentiable and the remainder termRnto tend to zero asntends to

infinity, i.e. limn→∞Rn= 0. In this case the infinite power series will represent the

function within the interval of convergence of the series.

Expandf(x)=sinxas a Maclaurin series, i.e. aboutx=0.

We must first verify that sinxmay indeed be represented by an infinite power series. It is
easily shown that thenth derivative off(x)isgivenby

f(n)(x)=sin

(

x+

nπ

2

)

.

Therefore the remainder after expandingf(x)asan(n−1)th-order polynomial about
x= 0 is given by

Rn(x)=

xn

n!

sin

(

ξ+

nπ

2

)

,

whereξlies in the range [0,x]. Since the modulus of the sine term is always less than or
equal to unity, we can write|Rn(x)|<|xn|/n!. For any particular value ofx,sayx=c,
Rn(c)→0asn→∞. Hence limn→∞Rn(x) = 0, and so sinxcan be represented by an
infinite Maclaurin series.
Evaluating the function and its derivatives atx=0weobtain

f(0) = sin 0 = 0,

f′(0) = sin(π/2) = 1,

f′′(0) = sinπ=0,

f′′′(0) = sin(3π/2) =− 1 ,

and so on. Therefore, the Maclaurin series expansion of sinxis given by

sinx=x−

x^3

3!

+

x^5

5!

−···.

Note that, as expected, since sinxis an odd function, its power series expansion contains
only odd powers ofx.