| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Infinite series convergence and sum |
| Difficulty | Standard +0.3 Part (a) is a routine limit requiring dividing numerator and denominator by a. Part (b) involves standard integration of rational functions using logarithms and evaluating an improper integral. Both are textbook exercises for FP3 with straightforward techniques and no novel insight required, making this slightly easier than average. |
| Spec | 4.08c Improper integrals: infinite limits or discontinuous integrands8.01c Sequence behaviour: periodic, convergent, divergent, oscillating, monotonic8.01d Sequence limits: limit of nth term as n tends to infinity, steady-states |
5
\begin{enumerate}[label=(\alph*)]
\item Show that $\lim _ { a \rightarrow \infty } \left( \frac { 3 a + 2 } { 2 a + 3 } \right) = \frac { 3 } { 2 }$.
\item Evaluate $\int _ { 1 } ^ { \infty } \left( \frac { 3 } { 3 x + 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x$, giving your answer in the form $\ln k$, where $k$ is a rational number.\\
(5 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2006 Q5 [7]}}