Find the first three terms, in ascending powers of xxx, of the binomial expansion of 19+x\frac{1}{\sqrt{9 + x}}9+x1
Hence, find the first three terms of the binomial expansion of 19−x2\frac{1}{\sqrt{9 - x^2}}9−x21
Using your answer to part (b), find an approximation for ∫0119−x2 dx\int_{0}^{1} \frac{1}{\sqrt{9 - x^2}} \, dx∫019−x21dx, giving your answer to seven decimal places.
Sarah, a student, decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used. Explain clearly whether Sarah's approximation will be an overestimate, an underestimate, or if it is impossible to tell.
Sarah goes on to use the expansion from part (b) to find an approximation for ∫0419−x2 dx\int_{0}^{4} \frac{1}{\sqrt{9 - x^2}} \, dx∫049−x21dx. Explain why Sarah's approximation is invalid.
Practise AQA A Level Maths 1.7 D: Sequences and series with exam-style questions for A Level Maths. 267 questions covering 1.7.1 Binomial expansion, 1.7.2 Types of sequence (A-level only), 1.7.3 Sigma notation (A-level only), 1.7.4 Arithmetic sequences and series (A-level only), 1.7.5 Geometric sequences and series (A-level only), and 1.7.6 Sequences and series in modelling (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
