| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 6 |
| Topic | Taylor series |
| Type | Use series to approximate numerical value |
| Difficulty | Standard +0.8 This is a Further Maths question requiring knowledge of Maclaurin series and series manipulation. Part (i) is straightforward substitution, part (ii) requires understanding convergence conditions, but part (iii) demands the insight to recognize the given series as ln(1 + 3x²) evaluated at a specific x value, then solve for that value—this non-trivial connection between abstract series and the expansion elevates it above routine exercises. |
| 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=(\roman*)]
\item Using the Maclaurin series for $\ln(1 + x)$, find the first four terms in the series expansion for $\ln(1 + 3x^2)$. [2]
\item Find the range of $x$ for which the expansion is valid. [1]
\item Find the exact value of the series
$$\frac{3^1}{2 \times 2^2} - \frac{3^2}{3 \times 2^4} + \frac{3^3}{4 \times 2^6} - \frac{3^4}{5 \times 2^8} + \ldots$$ [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q9 [6]}}