Understanding Engineering Mathematics

integers are subtracted fromnandn−r+1 will never become zero becauseris an integer

whereasnis not. So the expansion becomes an infinite series. In such cases it is called an

infinite power seriesinx. Such series can be used to express functions in a convenient

form for calculation, and we return to them in Chapter 14. For now, just accept the formal

generalisation to non-integer values ofn. It can be shown that the resulting infinite series

convergesif|x|< 1 ,divergesif|x|>1andif|x|=1 it may diverge or converge.

Example

1

1 −x

=( 1 −x)−^1 = 1 +x+x^2 +x^3 +··· provided|x|< 1

In this case it is obvious that the series will diverge for|x|≥1.

This series is a standard result of such importance that you should memorise it. This is

not too difficult – it is in fact the sum of an infinite GP with common ratiox.Itisuseful

in making approximations such as

1

1 −x

 1 +xforxvery small

Solution to review question 3.1.11

Applying the binomial expansion

( 1 +x)n= 1 +nx+···+

n(n− 1 )

2!

x^2 +

n(n− 1 )(n− 2 )

3!

x^3

+···+

n(n− 1 )...(n−r+ 1 )

r!

xr+···

we have

( 1 + 3 x)−^2 = 1 +(− 2 )( 3 x)+

(− 2 )(− 3 )

2!

( 3 x)^2

+

(− 2 )(− 3 )(− 4 )

3!

( 3 x)^3 +···

Note the careful treatment of the signs and the 3x– it is usually a good

policy to keep such things in brackets until the final stages of the calcula-

tion. Tidying up the results we find for the first four terms

( 1 + 3 x)−^2 = 1 − 6 x+ 27 x^2 − 108 x^3 +···

3.3 Reinforcement

3.3.1 Definition of a function

➤➤

88 90

➤

A.Find the values of the following functions at the points (a)−1, (b) 0, (c) 3 (d)−^13.

(i) f(x)= 2 x (ii) f(x)= 3 x^2 − 1