| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | February |
| Marks | 6 |
| Topic | Taylor series |
| Type | Maclaurin series of shifted function |
| Difficulty | Moderate -0.3 Part (a) is pure recall of a standard series. Part (b) requires substitution and expansion of e^(x³-1) = e^(-1)·e^(x³), which involves multiplying by a constant and recognizing that only the first term of e^x contributes to terms up to x³. This is a routine Further Maths technique with straightforward algebra, slightly easier than average due to the mechanical nature and limited algebraic manipulation required. |
| 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 Write down the Maclaurin series of $e^x$, in ascending power of $x$, up to and including the term in $x^3$ [1]
\item Hence, without differentiating, determine the Maclaurin series of
$$e^{(x^3-1)}$$
in ascending powers of $x$, up to and including the term in $x^3$, giving each coefficient in simplest form. [5]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q13 [6]}}