Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

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+


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+


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.

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