Mathematical Methods for Physics and Engineering : A Comprehensive Guide

4.6 TAYLOR SERIES

We may follow a similar procedure to obtain a Taylor series about an arbitrary

pointx=a.

Expandf(x)=cosxas a Taylor series aboutx=π/ 3.

As in the above example, it is easily shown that thenth derivative off(x)isgivenby

f(n)(x)=cos

(

x+

nπ

2

)

.

Therefore the remainder after expandingf(x)asan(n−1)th-order polynomial about
x=π/3isgivenby

Rn(x)=

(x−π/3)n

n!

cos

(

ξ+

nπ

2

)

,

whereξlies in the range [π/ 3 ,x]. The modulus of the cosine term is always less than or
equal to unity, and so|Rn(x)|<|(x−π/3)n|/n!. As in the previous example, limn→∞Rn(x)=
0 for any particular value ofx,andsocosxcan be represented by an infinite Taylor series
aboutx=π/3.
Evaluating the function and its derivatives atx=π/3weobtain
f(π/3) = cos(π/3) = 1/ 2 ,
f′(π/3) = cos(5π/6) =−

√

3 / 2 ,

f′′(π/3) = cos(4π/3) =− 1 / 2 ,

and so on. Thus the Taylor series expansion of cosxaboutx=π/3isgivenby

cosx=

1

2

−

√

3

2

(

x−π/ 3

)

−

1

2

(

x−π/ 3

) 2

2!

+···.

4.6.2 Approximation errors in Taylor series

In the previous subsection we saw how to represent a functionf(x) by an infinite

power series, which is exactly equal tof(x) for allxwithin the interval of

convergence of the series. However, in physical problems we usually do not want

to have to sum an infinite number of terms, but prefer to use only a finite number

of terms in the Taylor series toapproximatethe function in some given range

ofx. In this case it is desirable to know what is the maximum possible error

associated with the approximation.

As given in (4.18), a functionf(x) can be represented by a finite (n−1)th-order

power series together with a remainder term such that

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),

where

Rn(x)=

(x−a)n

n!

f(n)(ξ)

andξlies in the range [a, x].Rn(x) is the remainder term, and represents the error

in approximatingf(x)bytheabove(n−1)th-order power series. Since the exact