4PM1/02R Jun 2024, Question 2: Binomial Series for Any Rational Power
Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 5 marks
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The question, in short
Use binomial expansion on 2 divided by the square root of 1 + 3x, giving terms up to x³ with coefficients as simplified fractions, then give the x values for which this expansion holds.
Summarised in our own words. Use the official paper for the exact wording and any diagrams.
Part (a) · 4 marks
Write the binomial series for 2/√(1 + 3x) in increasing powers of x, stopping at the x³ term, with each coefficient reduced to a simple fraction.
- 12/√(1 + 3x) = 2(1 + 3x)^(−1/2)
- 2= 2[1 + (−1/2)(3x) + (−1/2)(−3/2)(3x)²/2! + (−1/2)(−3/2)(−5/2)(3x)³/3!]
- 3= 2[1 − (3/2)x + (27/8)x² − (135/16)x³]
- 4= 2 − 3x + (27/4)x² − (135/8)x³
Answer2 − 3x + (27/4)x² − (135/8)x³
Part (b) · 1 mark
Give the interval of x values over which the series from part a converges.
- 1Valid when |3x| < 1
- 2−1/3 < x < 1/3
Answer−1/3 < x < 1/3
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