| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 7 |
| Topic | Taylor series |
| Type | Maclaurin series for ln(exponential expressions) |
| Difficulty | Challenging +1.3 This is a Further Maths question requiring Maclaurin series via differentiation (involving chain rule with ln and cos), then solving a transcendental equation and manipulating to obtain a specific approximation for π. Part (a) requires careful differentiation and evaluation at x=0; part (b) needs insight to recognize cos x = -1/2 gives the root and connect this to the series approximation. More demanding than standard C3/C4 work but follows established FM techniques without requiring exceptional creativity. |
| Spec | 4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
\begin{enumerate}[label=(\alph*)]
\item Use differentiation to find the first two non-zero terms of the Maclaurin expansion of $\ln\left(\frac{1}{2} + \cos x\right)$.
[4]
\item By considering the root of the equation $\ln\left(\frac{1}{2} + \cos x\right) = 0$ deduce that $\pi \approx 3\sqrt{3 \ln\left(\frac{3}{2}\right)}$.
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q14 [7]}}