CAIE A-Level Further Mathematics 2.3.3 Maclaurin Series

Practise deriving the required number of Maclaurin terms from successive derivatives or an MF19 expansion, keeping exact constants and factorial denominators.

Exam points

find the required derivatives and evaluate each at zero before inserting factorial denominators
substitute into a standard MF19 expansion far enough to retain every requested power after simplification
keep logarithmic or other constants exact when matching the series to the form stated in the question

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

1

B1

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

1

M1

Values of derivatives at 0

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

1

M1

Use Maclaurin's series

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

1

M1

In terms of ln

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

1

A1

CAO

5

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2.3.3—Maclaurin series question 1