Maclaurin seriesEdexcel A-Level Further Maths: Subtopic test

1. 1
   The function $\mathrm{f}(x)=\mathrm{e}^{2x}$ is expanded as a Maclaurin series.

   (a)

   Find the coefficient of $x^3$ in the series.

   [1 mark]

   • A$8$
   • B$\frac23$
   • C$\frac43$
   • D$\frac13$

   (b)

   Which expression is the coefficient of $x^r$ in the series?

   [1 mark]

   • A$\frac{2}{r!}$
   • B$\frac{2^r}{r}$
   • C$\frac{2r}{r!}$
   • D$\frac{2^r}{r!}$

   (c)

   Use the first four terms of the series with $x=0.05$ to estimate the value of $\mathrm{e}^{0.1}$, giving your answer to 5 decimal places.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\mathrm{g}(x)=\ln(1+3x)$.

   (a)

   Find the coefficient of $x^2$ in the Maclaurin series of $\mathrm{g}(x)$.

   [1 mark]

   • A$\frac92$
   • B$-\frac92$
   • C$-\frac32$
   • D$-9$

   (b)

   For which values of $x$ is the series for $\mathrm{g}(x)$ valid?

   [1 mark]

   • A$-\frac13<x\leq\frac13$
   • B$-1<x\leq1$
   • C$-3<x\leq3$
   • D$-\frac13\leq x<\frac13$

   (c)

   Find the first three non-zero terms of the Maclaurin series of $\mathrm{g}(x)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $\mathrm{f}(x)=\mathrm{e}^x\sin x$.

   (a)

   Use the standard series for $\mathrm{e}^x$ and $\sin x$ to find the series expansion of $\mathrm{f}(x)$ up to and including the term in $x^3$.

   [3 marks]

   (b)

   Use your series to find an approximation, to 3 decimal places, for $\int_0^{0.5}\mathrm{e}^x\sin x\,\mathrm{d}x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two functions are defined by $\mathrm{f}(x)=\ln(1+\sin x)$ and $\mathrm{g}(x)=\ln\left(\frac{1+x}{1-x}\right)$.

   (a)

   Find the Maclaurin series of $\mathrm{f}(x)$ up to and including the term in $x^3$, by repeated differentiation.

   [6 marks]

   (b)

   (i) Write down the Maclaurin series of $\ln(1-x)$ up to and including the term in $x^3$.
   (ii) Hence show that the series for $\mathrm{g}(x)$ begins $2x+\frac23x^3$.
   (iii) State the range of values of $x$ for which this series is valid, and use $x=\frac13$ with the two terms in (ii) to estimate $\ln2$ to 3 decimal places.

   [6 marks]

   Total for question 4: 12 marks

End of questions