| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Partial fractions then method of differences |
| Difficulty | Moderate -0.3 This is a standard three-part question on partial fractions and telescoping series. Part (i) is routine A-level partial fractions, part (ii) applies the standard method of differences technique with clear cancellation, and part (iii) is a straightforward limit as nāā. While it requires multiple techniques, each step follows a well-practiced procedure with no novel insight needed, making it slightly easier than average. |
| Spec | 4.06b Method of differences: telescoping series |
\begin{enumerate}[label=(\roman*)]
\item Express $\frac{3}{(3r-1)(3r+2)}$ in partial fractions. [2]
\item Using the method of differences, prove that $\sum_{r=1}^{n} \frac{3}{(3r-1)(3r+2)} = \frac{1}{2} - \frac{1}{3n+2}$. [2]
\item Deduce the value of $\sum_{r=1}^{\infty} \frac{1}{(3r-1)(3r+2)}$. [1]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2018 Q1 [5]}}