1. Further binomial
Series
expansion

2. Series
KUS objectives
BAT review the binomial expansion and extend to expansion of a negative or fractional powers using a series expansion of (1 + x)n
Starter:

3. WB 1 (review)
a) find 𝟏+𝒙 𝟒
Always start by writing out the general form
𝒂+𝒃 𝒏 = 𝒂 𝒏 + 𝒏 𝟏 𝒂 𝒏−𝟏 𝒃+ 𝒏 𝟐 𝒂 𝒏−𝟐 𝒃 𝟐 + 𝒏 𝟑 𝒂 𝒏−𝟑 𝒃 𝟑 + …+ 𝒏 𝒏 𝒃 𝒏
Work out the fractions
1+𝑥 4 =1+4𝑥 +6 𝑥 2 +4 𝑥 3 + 𝑥 4
Every term after this one will contain a (0) so can be ignored
The expansion is finite and exact

4. WB 1b (review)
b) find 𝟏−𝟐𝒙 𝟑
We use the substitution x → -2x
𝒂+𝒃 𝒏 = 𝒂 𝒏 + 𝒏 𝟏 𝒂 𝒏−𝟏 𝒃+ 𝒏 𝟐 𝒂 𝒏−𝟐 𝒃 𝟐 + 𝒏 𝟑 𝒂 𝒏−𝟑 𝒃 𝟑 + …+ 𝒏 𝒏 𝒃 𝒏
It is VERY important to put brackets around the x parts
Work out the fractions
1−2𝑥 3 =1−6𝑥 +12 𝑥 2 −8 𝑥 …

5. 𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )
There is a shortened version of the expansion when one of the terms is 1
Whatever power 1 is raised to, it will be 1, and can therefore be ignored
The coefficients give values from Pascal’s triangle.
 For example, if n was 4…
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )

6. WB2a Find the first 4 terms of the Binomial expansion of
a) (1 + 2x)5 b) 2−𝑥 c) 2−𝑥 𝑑) 𝑥 5
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )
1+2𝑥 5 =1+5 2𝑥 + 5(4) 2 4 𝑥 (4)(3) 6 8 𝑥 3
Put the numbers in
Work out the fractions
=1+10𝑥 +40 𝑥 𝑥 …

7. WB2b Find the first 4 terms of the Binomial expansion of 𝑏) 2−𝑥 6
2−𝑥 6 = × 1− 1 2 𝑥 6 = − 𝑥 2 6
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )
1− 𝑥 =1+6 − 𝑥 (5) 2 𝑥 (5)(4) 6 − 𝑥 3 8
Put the numbers in
Work out the fractions
=1 −3𝑥 𝑥 − 5 2 𝑥 …
Useful for A2
Remember to multiply by 64!
64 1− 𝑥 =64−192𝑥+240 𝑥 2 −160 𝑥 3 +…

8. WB2c Find the first 4 terms of the Binomial expansion of 𝑐) 2−2𝑥 4
2−2𝑥 4 = × 1−𝑥 4 = −𝑥 4
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )
Put the numbers in
1−𝑥 4 =1+4 −𝑥 + 4(3) 2 𝑥 (3)(2) 6 − 𝑥 3
Work out the fractions
=1 −4𝑥 𝑥 −4 𝑥 …
Useful for A2
Remember to multiply by 16!
16 1−𝑥 4 =16−64𝑥+96 𝑥 2 −64 𝑥 3 +…

9. WB2d Find the first 4 terms of the Binomial expansion of 𝑏) 3+9𝑥 5
3+9𝑥 5 = × 1+3𝑥 5 =
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )
Put the numbers in
1+3𝑥 5 =1+5 3𝑥 + 5(4) 2 9 𝑥 (4)(3) 𝑥 3
Work out the fractions
=1+15𝑥 𝑥 𝑥 3 +…
Useful for A2
Remember to multiply by 243
𝑥 5 = 𝑥 𝑥 2 − 𝑥 3 +…

10. Negative or fraction coefficients give INFINITE series when expanded
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 𝒙 𝒓 +…
Goes on forever with increasing powers of x
SO we usually give the expansion to a set number of terms
Usually up to 𝑥 3 or 𝑥 4
We can only use the above expansion
DO NOT use the expansion for 𝒂+𝒃 𝒏

11. Rewrite this as a power of x first
WB 3 find the binomial expansion of up to the term in x3 a) 𝟏 𝟏+𝒙 𝐛) 𝟏−𝟑𝒙
Rewrite this as a power of x first
𝟏 𝟏+𝒙 = 𝟏+𝒙 −𝟏
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )

12. Rewrite this as a power of x first
WB 3b find the binomial expansion of 𝐛) 𝟏−𝟑𝒙
Rewrite this as a power of x first
𝟏−𝟑𝒙 = 𝟏+𝒙 −𝟏
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )

13. Explain why x=2 is not valid by substituting into your answer for (a)
WB 4a find the binomial expansion of the following and state the values of x for which each is valid 𝒂) 𝟏−𝒙 𝟏 𝟑 𝒃) 𝟏−𝟑𝒙
Explain why x=2 is not valid by substituting into your answer for (a)
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )
Imagine we substitute x = 2 into the expansion
The values fluctuate (easier to see as decimals)
 The result is that the sequence will not converge and hence for x = 2, the expansion is not valid

14. WB 4a (cont) find the binomial expansion of the following and state the values of x for which each is valid 𝒂) 𝟏−𝒙 𝟏 𝟑 𝒃) 𝟏 𝟏+𝟒𝒙 𝟐
Imagine we substitute x = 0.5 into the expansion
The values continuously get smaller
 This means the sequence will converge (like an infinite series) and hence for x = 0.5, the sequence IS valid…
How do we work out for what set of values x is valid?
The reason an expansion diverges or converges is down to the x term…
If the term is bigger than 1 or less than -1, squaring/cubing etc will accelerate the size of the term, diverging the sequence
If the term is between 1 and -1, squaring and cubing cause the terms to become increasingly small, so the sum of the sequence will converge, and be valid
Write using Modulus
The expansion is valid when the modulus value of x is less than 1

15. WB 4b find the binomial expansion of the following and state the values of x for which each is valid 𝒂) 𝟏−𝒙 𝟏 𝟑 𝒃) 𝟏 𝟏+𝟒𝒙 𝟐
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )
The ‘x’ term is 4x…

16. WB 5 Find the binomial expansion of 𝟏−𝟐𝒙
and by using 𝒙=𝟎.𝟎𝟏 find an estimate for 𝟐
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )
Put x = 0.01
RHS = 1 – 0.01 – – =

17. Check with calculator – this is accurate to 6 decimal places
WB 6 use a binomial expansion to find an approximation to 𝟏𝟎𝟐 give your answer to 5 decimal places
102 = =
An appropriate expansion is 1+2𝑥 with 𝑥=0.01
𝟏+𝒙 𝒏 =𝟏+𝒏𝒙+ 𝒏 𝒏−𝟏 𝟐! 𝒙 𝟐 + 𝒏 𝒏−𝟏 (𝒏−𝟐) 𝟑! 𝒙 𝟑 + …+ 𝒏 𝑪 𝒓 ( 𝒙 𝒓 )
Put the numbers in
1+2𝑥 1/2 = 𝑥 − 𝑥 − − 𝑥 3
Work out the fractions
=1 + 𝑥 − 𝑥 𝑥 3
Substitute x=0.01
= − =
Remember to x10
102 =
Check with calculator – this is accurate to 6 decimal places

18. Skills 2
HWK 2

19. One thing to improve is –
KUS objectives
BAT review the binomial expansion and extend to expansion of a negative or fractional powers
self-assess
One thing learned is –
One thing to improve is –

20. END