CAIE A-Level Further Math A2 2.3.3 Maclaurin Series Questions

Practise finding Maclaurin series expansions and using them for function approximations with Further Mathematics Paper 2 questions and mark schemes.

Exam points

Find and use Maclaurin series expansions for functions and related approximations.

[Maximum number: 5]

Hence find the first three terms in the Maclaurin's series for $\tanh ^{-1}\left(\frac{1-x}{2+x}\right)$ in the form

a \ln 3+b x+c x^{2},

where a, b and c are constants to be determined.

Differentiate again: $y^{\prime \prime}=\frac{2}{(2 x+1)^{2}}$

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Or differentiate result in (b): $(2 x+1) y^{\prime \prime}+2 y^{\prime}=0$ $y(0)=\tanh ^{-1}\left(\frac{1}{2}\right)$

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Values of derivatives at 0

$y^{\prime}(0)=-1 ; y^{\prime \prime}(0)=2$

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Use Maclaurin's series

$y=\frac{1}{2} \ln 3-x+2 \times \frac{x^{2}}{2}$

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In terms of ln

$=\frac{1}{2} \ln 3-x+x^{2}$

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CAO

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CAIE A-Level Further Math A2 2.3.3 Maclaurin Series Questions question 1