| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2022 |
| Session | February |
| Marks | 6 |
| Topic | Taylor series |
| Type | Use series to approximate numerical value |
| Difficulty | Challenging +1.2 This is a structured multi-part inequality problem requiring Maclaurin series (standard FM topic), algebraic manipulation with substitutions, and creative application to compare e^π vs π^e. While it requires some insight for the final part, the question provides heavy scaffolding through parts (a) and (b), making it more accessible than it initially appears. The techniques are standard for Further Maths, though the comparison in part (c) requires modest problem-solving to choose the right substitution. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item By using an appropriate Maclaurin series prove that if $x > 0$ then $e^x > 1 + x$. [2]
\item Hence, by using a suitable substitution, deduce that $e^t > et$ for $t > 1$. [1]
\item Using the inequality in part (b), and by making a suitable choice for $t$, determine which is greater, $e^{\pi}$ or $\pi^e$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q12 [6]}}