| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 7 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Partial fractions then method of differences |
| Difficulty | Standard +0.3 This is a standard telescoping series question requiring partial fractions decomposition of 1/(r²+3r+2) = 1/((r+1)(r+2)), followed by recognizing the telescoping sum pattern. The technique is well-practiced in Further Maths courses, and part (b) is trivial once (a) is complete. Slightly easier than average due to its routine nature and clear structure. |
| Spec | 4.05c Partial fractions: extended to quadratic denominators4.06b Method of differences: telescoping series |
In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item By using partial fractions show that $\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2} = \frac{1}{2} - \frac{1}{n+2}$. [5]
\item Hence determine the value of $\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2}$. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q1 [7]}}