Higher Engineering Mathematics, Sixth Edition

The binomial series 59

Hence

(a+x)^7 =a^7 + 7 a^6 x+ 21 a^5 x^2 + 35 a^4 x^3

+ 35 a^3 x^4 + 21 a^2 x^5 + 7 ax^6 +x^7

Problem 2. Determine, using Pascal’s triangle

method, the expansion of( 2 p− 3 q)^5.

Comparing( 2 p− 3 q)^5 with(a+x)^5 shows that
a= 2 pandx=− 3 q.
Using Pascal’s triangle method:

(a+x)^5 =a^5 + 5 a^4 x+ 10 a^3 x^2 + 10 a^2 x^3 +···

Hence

( 2 p− 3 q)^5 =( 2 p)^5 + 5 ( 2 p)^4 (− 3 q)

+ 10 ( 2 p)^3 (− 3 q)^2

+ 10 ( 2 p)^2 (− 3 q)^3

+ 5 ( 2 p)(− 3 q)^4 +(− 3 q)^5

i.e.(2p− 3 q)^5 = 32 p^5 − 240 p^4 q+ 720 p^3 q^2

− 1080 p^2 q^3 + 810 pq^4 − 243 q^5

Now try the following exercise

Exercise 28 Further problemson Pascal’s

triangle

1. Use Pascal’s triangle to expand(x−y)^7.
   [
   x^7 − 7 x^6 y+ 21 x^5 y^2 − 35 x^4 y^3
   
   
   • 35 x^3 y^4 − 21 x^2 y^5 + 7 xy^6 −y^7
   
   
   
   

]

2. Expand( 2 a+ 3 b)^5 using Pascal’s triangle.
   [
   32 a^5 + 240 a^4 b+ 720 a^3 b^2
   
   
   • 1080 a^2 b^3 + 810 ab^4 + 243 b^5
   
   
   
   

]

7.2 The binomial series

Thebinomial seriesorbinomial theoremis a formula
for raising a binomial expression to any power without
lengthy multiplication.The general binomial expansion

of(a+x)nis given by:

(a+x)n=an+nan−^1 x+

n(n−1)

2!

an−^2 x^2

+

n(n−1)(n−2)

3!

an−^3 x^3

+ ···

where 3! denotes 3× 2 ×1 and is termed ‘factorial 3’.

With the binomial theoremnmay be a fraction, a

decimal fraction or a positive or negative integer.

Whennis a positive integer, the series is finite, i.e.,

it comes to an end; whennisanegativeinteger,ora

fraction, the series is infinite.

In the general expansion of(a+x)nit is noted that the

4th term is:

n(n− 1 )(n− 2 )

3!

an−^3 x^3. The number 3 is

very evident in this expression.

For any term in a binomial expansion, say ther’th

term,(r− 1 )is very evident. It may therefore be rea-

soned thatther’th term of the expansion(a+x)nis:

n(n−1)(n−2)...to (r−1) terms

(r−1)!

an−(r−1)xr−^1

Ifa=1 in the binomial expansion of(a+x)nthen:

(1+x)n= 1 +nx+

n(n−1)

2!

x^2

+

n(n−1)(n−2)

3!

x^3 +···

which is valid for− 1 <x<1.

Whenxis small compared with 1 then:

( 1 +x)n≈ 1 +nx

7.3 Worked problems on the binomial series

Problem 3. Use the binomial series to determine

the expansion of( 2 +x)^7.

The binomial expansion is given by:

(a+x)n=an+nan−^1 x+

n(n− 1 )

2!

an−^2 x^2

+

n(n− 1 )(n− 2 )

3!

an−^3 x^3 +···