Further binomial Series expansion

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:

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

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 3 +…

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 )

WB2a Find the first 4 terms of the Binomial expansion of a) (1 + 2x)5 b) 2−𝑥 6 c) 2−𝑥 6 𝑑) 3+9𝑥 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 2 + 5(4)(3) 6 8 𝑥 3 Put the numbers in Work out the fractions =1+10𝑥 +40 𝑥 2 +80 𝑥 3 +…

WB2b Find the first 4 terms of the Binomial expansion of 𝑏) 2−𝑥 6 2−𝑥 6 = 2 6 × 1− 1 2 𝑥 6 = 64 1− 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 2 6 =1+6 − 𝑥 2 + 6(5) 2 𝑥 2 4 + 6(5)(4) 6 − 𝑥 3 8 Put the numbers in Work out the fractions =1 −3𝑥 + 15 4 𝑥 2 − 5 2 𝑥 3 +… Useful for A2 Remember to multiply by 64! 64 1− 𝑥 2 6 =64−192𝑥+240 𝑥 2 −160 𝑥 3 +…

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

WB2d Find the first 4 terms of the Binomial expansion of 𝑏) 3+9𝑥 5 3+9𝑥 5 = 3 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 2 + 5(4)(3) 6 27 𝑥 3 Work out the fractions =1+15𝑥 + 90 𝑥 2 + 270 𝑥 3 +… Useful for A2 Remember to multiply by 243 243 1+3𝑥 5 =243+3645𝑥+21870 𝑥 2 −656610 𝑥 3 +…

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

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 )    

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 )    

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          

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  

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…    

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 – 0.00005 – 0.0000005 = 0.9899495    

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 = 100 1.02 =10 1.02 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 =1+ 1 2 2𝑥 + 1 2 − 1 2 2 4 𝑥 2 + 1 2 − 1 2 − 3 2 6 8 𝑥 3 Work out the fractions =1 + 𝑥 − 1 2 𝑥 2 + 1 2 𝑥 3 Substitute x=0.01 =1 + 0.01 − 1 2 0.01 2 + 1 2 0.01 3 =1.0099505 Remember to x10 102 =10.099505 Check with calculator – this is accurate to 6 decimal places

Skills 2 HWK 2

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 –

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