A2-Level Maths: Core 4 for Edexcel C4.3 Sequences and series These icons indicate that teacher’s notes or useful web addresses are available in the Notes Page. This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation. 1 of 26 © Boardworks Ltd 2006

Negative and fractional indices Binomial expansion for negative and fractional indices Approximations Use of partial fractions Examination-style question Contents 2 of 26 © Boardworks Ltd 2006

Binomial expansion Previously in the course we found that, when n is a positive whole number, This is a finite series with n + 1 terms. If n is negative or fractional then, provided that |x| < 1, the infinite series will converge towards (1 + x)n.

Binomial expansion In general, for negative and fractional n and |x| < 1, Expand up to the term in x4. Start by writing this as (1 + x)–1. The expansion is then: This is equal to (1 + x)–1 provided that |x| < 1.

Binomial expansion Expand up to the term in x3. Start by writing this as . Here x is replaced by 2x. This converges towards provided that |2x| < 1. That is when |x| < .

Binomial expansion When the first term in the bracket is not 1, we have to factorize it first. For example: Find the first four terms in the expansion of (3 – x)–2.

Binomial expansion Therefore This expansion is valid for < 1. That is, when |x| < 3. In general, if we are asked to expand an expression of the form (a + bx)n where n is negative or fractional we should start by writing this as: The corresponding binomial expansion will be valid for |x| < .

Binomial expansion Expand up to the term in x2 giving the range of values for which the expansion is valid.

Binomial expansion Therefore This expansion is valid for < 1. That is, when |x| < .

Contents Approximations Binomial expansion for negative and fractional indices Approximations Use of partial fractions Examination-style question Contents 10 of 26 © Boardworks Ltd 2006

Approximations In general, when the index is negative or fractional, we only have to find the first few terms in a binomial expansion. This is because, as long as x is defined within a valid range, the terms get very small as the series progresses. For example, it can be shown that: If x is equal to 0.1 we have: By only using the first few terms in an expansion we can therefore give a reasonable approximation.

Approximations

Approximations If we only expand up to the term in x it is called a linear approximation. For example: (for |x| < 1) If we expand up to the term in x2 it is called a quadratic approximation. For example: (for |x| < 1) Binomial expansions can be used to make numerical approximations by choosing suitable values for x. Write a quadratic approximation to and use this to find a rational approximation for .

Approximations (for |x| < 4) Let x = 1:

Approximations Expand up to the term in x2 and substitute x = to obtain a rational approximation for (for |x| < 1) When x = we have:

Approximations Therefore We can check the accuracy of this approximation using a calculator. Our approximation is therefore correct to 2 decimal places. If a greater degree of accuracy is required we can extend the expansion to include more terms.

Use of partial fractions Binomial expansion for negative and fractional indices Approximations Use of partial fractions Examination-style question Contents 17 of 26 © Boardworks Ltd 2006

Use of partial fractions We can use partial fractions to carry out more complex binomial expansions. For example, we can expand by expressing it in partial fractions as follows: Let Multiplying through by (x + 1)(x – 2) gives: When x = –1:

Use of partial fractions When x = 2: So We can now expand 2(1 + x)–1 and 3(–2 + x)–1 :

Use of partial fractions This is valid for |x| < 1. This expands to give: This is valid for |x| < 2.

Use of partial fractions We can now add the two expansions together: This is valid when both |x| < 1 and |x| < 2. –2 –1 1 2 From the number line we can see that both inequalities hold when |x| < 1.

Examination-style question Binomial expansion for negative and fractional indices Approximations Use of partial fractions Examination-style question Contents 22 of 26 © Boardworks Ltd 2006

Examination-style question The function f is given by Express f(x) in partial fractions. Expand f(x) as a series in ascending powers of x as far as the term in x2. State the set of values of x for which the expansion is valid. a) Let Multiplying through by (1 + 2x)(1 – x) gives:

Examination-style question When x = : When x = 1: So b)

Examination-style question Expanding 4(1 + 2x)–1 gives: Expanding 2(1 – x)–1 gives:

Examination-style question Therefore f(x) can be expanded as: c) The expansion of (1 + 2x)–1 is valid for |x| < . The expansion of (1 – x)–1 is valid for |x| < 1. The expansion of f(x) is therefore valid for |x| < .