| Exam Board | CAIE |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Rectangle bounds for infinite series |
| Difficulty | Challenging +1.2 This is a structured Further Maths question on using rectangles to bound integrals and series. Part (a) requires recognizing that rectangles give a lower bound for the integral, then algebraic manipulation. Parts (b) and (c) follow the same pattern with guidance. The technique is standard for Further Maths students studying series convergence, requiring careful bookkeeping but no novel insight. Slightly above average difficulty due to the algebraic manipulation and being Further Maths content. |
| Spec | 1.08g Integration as limit of sum: Riemann sums4.06a Summation formulae: sum of r, r^2, r^3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2^{-2} + \ldots + (N-1)^{-2} = \sum_{r=2}^{N-1} \frac{1}{r^2}\) | B1 | Forms sum of areas of rectangles; must be written explicitly, may be in summation form |
| \(\int_2^N x^{-2}\, dx = \left[-x^{-1}\right]_2^N = \frac{1}{2} - \frac{1}{N}\) | M1 A1 | Evaluates integral, correct limits |
| \(\sum_{r=1}^N r^{-2} > \int_2^N x^{-2}\, dx + 1 + \frac{1}{N^2}\) | M1 | Compares with integral correctly (using \(>\)) |
| \(\sum_{r=1}^N r^{-2} > \frac{3}{2} - \frac{1}{N} + \frac{1}{N^2}\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3^{-2} + 4^{-2} + \ldots + N^{-2}\) | M1* | Forms sum of areas of appropriate rectangles; allow addition of one extra rectangle or shift one left or right if limits adjusted accordingly |
| \(< \int_2^N x^{-2}\, dx = \left[-x^{-1}\right]_2^N = \frac{1}{2} - \frac{1}{N}\) | DM1 | Compares with integral, correct limits and correct inequality (\(<\)) |
| \(\sum_{r=1}^N r^{-2} < \frac{7}{4} - \frac{1}{N}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sum_{r=1}^{\infty} r^{-2} > \frac{3}{2}\) | B1 | Needs to clarify (using \(>\) or words) that value is lower bound |
| \(\sum_{r=1}^{\infty} r^{-2} < \frac{7}{4}\) | B1 FT | Needs to clarify (using \(<\) or words) that value is upper bound. Must have scored final A mark in 4(b). FT on fully correct alternative to part 4(b) |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2^{-2} + \ldots + (N-1)^{-2} = \sum_{r=2}^{N-1} \frac{1}{r^2}$ | B1 | Forms sum of areas of rectangles; must be written explicitly, may be in summation form |
| $\int_2^N x^{-2}\, dx = \left[-x^{-1}\right]_2^N = \frac{1}{2} - \frac{1}{N}$ | M1 A1 | Evaluates integral, correct limits |
| $\sum_{r=1}^N r^{-2} > \int_2^N x^{-2}\, dx + 1 + \frac{1}{N^2}$ | M1 | Compares with integral correctly (using $>$) |
| $\sum_{r=1}^N r^{-2} > \frac{3}{2} - \frac{1}{N} + \frac{1}{N^2}$ | A1 | AG |
---
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3^{-2} + 4^{-2} + \ldots + N^{-2}$ | M1* | Forms sum of areas of appropriate rectangles; allow addition of one extra rectangle or shift one left or right if limits adjusted accordingly |
| $< \int_2^N x^{-2}\, dx = \left[-x^{-1}\right]_2^N = \frac{1}{2} - \frac{1}{N}$ | DM1 | Compares with integral, correct limits and correct inequality ($<$) |
| $\sum_{r=1}^N r^{-2} < \frac{7}{4} - \frac{1}{N}$ | A1 | |
---
## Question 4(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sum_{r=1}^{\infty} r^{-2} > \frac{3}{2}$ | B1 | Needs to clarify (using $>$ or words) that value is lower bound |
| $\sum_{r=1}^{\infty} r^{-2} < \frac{7}{4}$ | B1 FT | Needs to clarify (using $<$ or words) that value is upper bound. Must have scored final A mark in 4(b). FT on fully correct alternative to part 4(b) |
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\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_408_1433_296_315}
The diagram shows the curve with equation $y = x ^ { - 2 }$ for $2 \leqslant x \leqslant N$ together with a set of ( $N - 2$ ) rectangles of unit width.
\begin{enumerate}[label=(\alph*)]
\item By considering the sum of the areas of these rectangles, show that
$$\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } } > \frac { 3 } { 2 } - \frac { 1 } { N } + \frac { 1 } { N ^ { 2 } }$$
\includegraphics[max width=\textwidth, alt={}, center]{27485e4a-cd34-43e3-aa92-767820a9f6f9-08_2718_35_141_2012}
\begin{center}
\end{center}
\item Use a similar method to find, in terms of $N$, an upper bound for $\sum _ { r = 1 } ^ { N } \frac { 1 } { r ^ { 2 } }$.
\item Deduce lower and upper bounds for $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 2 2024 Q4 [10]}}