| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2003 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Sum from n+1 to 2n or similar range |
| Difficulty | Challenging +1.2 This is a telescoping series problem requiring algebraic manipulation to find a closed form, followed by solving an inequality involving large powers. While it requires recognizing the telescoping pattern and careful bookkeeping of terms, the techniques are standard for Further Maths. The second part is computational rather than conceptually demanding. Harder than average A-level but routine for FP1. |
| Spec | 4.06b Method of differences: telescoping series8.01a Recurrence relations: general sequences, closed form and recurrence |
Given that
$$u_n = \frac{1}{n^2 - n + 1} - \frac{1}{n^2 + n + 1},$$
find $S_N = \sum_{n=N+1}^{2N} u_n$ in terms of $N$. [3]
Find a number $M$ such that $S_N < 10^{-20}$ for all $N > M$. [3]
\hfill \mbox{\textit{CAIE FP1 2003 Q2 [6]}}